Carter Subgroups of Finite Groups

c Allerton Press, Inc., 2009. ISSN 1055-1344, Siberian Advances in Mathematics, 2009, Vol. 19, No. 1, pp. 24–74. c E. P. Vdovin, 2008, published in Matematicheskie Trudy, 2008, Vol. 11, No. 2, pp. 20–106. Original Russian Text

Carter Subgroups of Finite Groups E. P. Vdovin1* 1

Sobolev Institute of Mathematics, Novosibirsk, 630090 Russia Received January 14, 2008

Abstract—It is proven that the Carter subgroups of a finite group are conjugate. A complete classification of the Carter subgroups in finite almost simple groups is also obtained. DOI: 10.3103/S1055134409010039 Key words: Carter subgroup, finite simple group, group of Lie type, linear algebraic group, semilinear group of Lie type, semilinear algebraic group, conjugated powers of an element

Contents 1

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Main results . . . . . . . . . . . . . . . Preliminary results . . . . . . . . . . . Proof of Theorem 2.1.4 . . . . . . . . Some properties of Carter subgroups

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25 26 27 29 32 33 33 34 35 35 37

Brief review of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost simple groups which are not minimal counter examples . . . . . . . . . . . . .

37 38 45 48

Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Translation of basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carter subgroups of special type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief review of the results . . . . . . . . . . . . . . . . . . . Carter subgroups of symplectic groups . . . . . . . . . . . Groups with triality automorphism . . . . . . . . . . . . . Classification theorem . . . . . . . . . . . . . . . . . . . . . Carter subgroups of order divisible by characteristic . . . Carter subgroups of order not divisible by characteristic

E-mail: [email protected]

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CARTER SUBGROUPS OF SEMILINEAR GROUPS

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SEMILINEAR GROUPS OF LIE TYPE

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CONJUGACY IN SIMPLE GROUPS

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General characteristic of the results . . . Notation and results from Group theory Linear algebraic groups . . . . . . . . . . Structure of finite groups of Lie type . . Known results . . . . . . . . . . . . . . .

CONJUGACY CRITERION FOR CARTER SUBGROUPS

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INTRODUCTION

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CARTER SUBGROUPS OF FINITE GROUPS 7 6

Carter subgroups of finite groups are conjugate . . . . . . . . . . . . . . . . . . . . . . .

25 66 66

EXISTENCE CRITERION

1 2 3

Brief review of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Classification of Carter subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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REFERENCES

1. INTRODUCTION The present paper is a slightly shorten version of the doctoral thesis “Carter subgroups of finite groups”. The results of the thesis were published in [33, 39, 42–45]. 1.1. General characteristic of the results . We recall that a subgroup of a finite group is called a Carter subgroup if it is nilpotent and self-normalizing. It is well known that any finite solvable group contains exactly one conjugacy class of Carter subgroups (see [3]). If a group is not assumed to be finite then Carter subgroups can be even nonisomorphic. Indeed, if N1 and N2 are two nonisomorphic nilpotent groups then they are Carter subgroups in their free product. On the other hand, a finite nonsolvable group may fail to contain Carter subgroups, the minimal counter example is the alternating group of degree 5. Although there is not a single example of a finite group containing nonconjugate Carter subgroups, the following problem due to R. Carter is known. Problem 1.1.1. (Conjugacy Problem) Are the Carter subgroups of a finite group conjugate? This problem for several classes of finite groups close to simple was studied by many authors. For example, L. Di Martino and M. C. Tamburini classified the Carter subgroups in symmetric and alternating groups (see [14]) and also in every group G such that SLn (q) ≤ G ≤ GLn (q) (see [16]). The case G = GLn (q) was considered by the same authors earlier in [15] and, independently, by N. A. Vavilov in [41]. For symplectic groups Sp2n (q), general unitary groups GUn (q), and finally, for general orthogonal groups GO± n (q) with odd q, the classification of the Carter subgroups was obtained by L. Di Martino, A. E. Zalessky, and M. C. Tamburini (see [17]). For some sporadic simple groups, Carter subgroups were found in [12]. In the nonsolvable groups mentioned above, Carter subgroups coinside with the normalizers of Sylow 2-subgroups, and hence, are conjugate. A finite group G is called a minimal counter example to Conjugacy Problem or a minimal counter example for brevity if G contains nonconjugate Carter subgroups, but in every group H, with |H| < |G|, the Carter subgroups are conjugate. In [11], F. Dalla Volta, A. Lucchini, and M. C. Tamburini proved that a minimal counter example must be almost simple. This result allows to use the classification of finite simple groups to solve Conjugacy Problem. Note that the using of the above-mentioned result by F. Dalla Volta, A. Lucchini, and M. C. Tamburini for the classification of Carter subgroups in almost simple groups essentially depends on the classification of finite simple groups. Indeed, in order to use the inductive hypothesis that the Carter subgroups in every proper subgroup of a minimal counter example are conjugate, one needs to know that all almost simple groups of order less than the order of a minimal counter example are found. To avoid using the classification of finite simple groups, we strengthen the result from [11] proving that if Carter subgroups are conjugate in the group of induced automorphisms of every non-Abelian composition factor then they are conjugate in the group. For inductive description of the Carter subgroups in almost simple groups, one needs to know homomorphic images of Carter subgroups and intersections of Carter subgroups with normal subgroups, i. e., to answer the following questions: SIBERIAN ADVANCES IN MATHEMATICS

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Problem 1.1.2. Is a homomorphic image of a Carter subgroup again a Carter subgroup? Problem 1.1.3. Is the intersection of a Carter subgroup with a normal subgroup again a Carter subgroup (of the normal subgroup)? The first problem is closely connected with Conjugacy Problem. Namely, if Conjugacy Problem has an affirmative answer then the first problem also has an affirmative answer. So we will solve both of these problems by considering Carter subgroups in almost simple groups. It is easy to see that the second problem has a negative answer. Indeed, consider a solvable group Sym3 and its normal subgroup of index 2, i. e., the alternating group Alt3 . Then a Carter subgroup of Sym3 is a Sylow 2-subgroup, while a Carter subgroup of Alt3 is a Sylow 3-subgroup. Thus, in the paper, some properties of Carter subgroups in a group and some of its normal subgroups are studied. The present paper is divided into six Sections including Introduction. In Introduction, we give general results of the paper and some necessary definitions and results as well. In Section 2, we prove that Carter subgroups of a finite group are conjugate if they are conjugate in the group of induced automorphisms of every its non-Abelian composition factor, thereby we strengthen the results by F. Dalla Volta, A. Lucchini, and M. C. Tamburini. In Section 2, we also study some properties of Carter subgroups. In Section 3, we consider the problem of conjugacy for elements of prime order in finite groups of Lie type. At the end of Section 3, using the results on conjugacy, we obtain the classification of Carter subgroups in a broad class of almost simple groups. In Section 4, we introduce the notion of semilinear groups of Lie type and the corresponding semilinear algebraic groups and transfer the results to the normalizers of p-subgroups and to the centralizers of semisimple elements in groups of Lie type. We also obtain some additional results on the conjugacy of elements of prime order in these groups. In Section 5, we complete the classification of Carter subgroups in almost simple groups and prove that the Carter subgroups of almost simple groups are conjugate. As a corollary, we obtain an affirmative answer to Conjugacy Problem and prove that a homomorphic image of a Carter subgroup is a Carter subgroup. In Section 6, we study the problem of existence of a Carter subgroup in a finite group, give a criterion of the existence and construct an example showing that the property of containing a Carter subgroup is not preserved under the extensions. Moreover, in the last Subsection of this Section, we give tables with the classification of Carter subgroups in almost simple groups. 1.2. Notation and results from Group theory. We will use the standard notation. If G is a group then H ≤ G and H E G mean that H is a subgroup and a normal subgroup of G respectively. By |G : H| we denote the index of H in G, NG (H) is the normalizer of H in G. If H is normal in G then by G/H we denote the factor group of G by H. If M is a subset of G then hM i denotes the subgroup generated by M , and |M | denotes the cardinality of M (or the order of an element if there is an element instead of a set). By CG (M ) we denote the centralizer of M in G, and by Z(G) we denote the center of G. The conjugation of x by an element y in G is written as xy = y −1 xy (y x = yxy −1 ), and by [x, y] = x−1 xy we denote the commutator of x, y. The symbol [A, B] means the mutual commutant of subgroups A and B of G. For groups A and B, the expressions A × B, A ◦ B, and A ⋌ B (or B ⋋ A) mean direct, central, and semidirect products of A and B, with B normal, respectively. If S ≤ Symn and G is a group then G ≀ S denoted the permutation wreath product. If A and B are subgroups of G such that A E B, then the factor group B/A is called a section of G. The Fitting subgroup of G is denoted by F (G), the generalized Fitting subgroup is denoted by F ∗ (G). The set of Sylow p-subgroups of a finite group G will be denoted by Sylp (G). If ϕ is a homomorphism of G and g is an element of G then Gϕ and gϕ are the images of G and g under ϕ respectively. By Gϕ we denote the set of stable points of G under the endomorphism ϕ. By Aut(G), Out(G), and Inn(G) we denote the group of all automorphisms, the group of outer automorphisms, and the group of inner automorphisms of G respectively. If G is a group then we denote by PG the factor group G/Z(G). The isomorphism PG ≃ Inn(G) is known, in particular, if Z(G) is trivial then G ≃ Inn(G) and we may assume that G ≤ Aut(G). A finite group G is said to be almost simple if there exists a finite group S, with S ≤ G ≤ Aut(S), i. e., F ∗ (G) = S is a simple group. For every positive integer t, we denote by Zt a cyclic group of order t. SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009

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If π is a set of primes, then by π ′ we denote its complement in the set of all primes. For every positive integer n, by π(n) we denote the set of prime divisors of n, and by nπ we denote the maximal divisor of n such that π(nπ ) ⊆ π. As usual, we denote by Oπ (G) the maximal normal π-subgroup of G, and ′ we denote by Oπ (G) the subgroup generated by all π-elements of G. If π = {2}′ is a set of all odd primes, then Oπ (G) = O2′ (G) is denoted by O(G). If g ∈ G, then by gπ we denote the π-part of g, i. e., gπ = g|g|π′ . Let G be a group, let A, B, and H be subgroups of G, and let B be normal in A. Then NH (A/B) = NH (A) ∩ NH (B). If x ∈ NH (A/B), then x induces the automorphism Ba 7→ Bx−1 ax of A/B. Thus there exists a homomorphism of NH (A/B) into Aut(A/B). The image of this homomorphism is denoted by AutH (A/B) and is called a group of induced automorphisms of H on the section A/B. In particular, if S = A/B is a composition factor of G then, for each subgroup H ≤ G, the group AutH (S) = AutH (A/B) is well defined. Note that the structure of AutH (S) depends on the choice of a composition series. If A, H are subgroups of G then, by definition, AutH (A) = AutH (A/{e}). 1.3. Linear algebraic groups . The necessary information about the structure and properties of linear algebraic groups can be found in [25]. Since we consider only linear algebraic groups, we will omit the word “linear” for brevity. 0

If G is an algebraic group then by G we denote the unit component of G. An algebraic group is called semisimple if its radical R(G) is trivial, and an algebraic group is called reductive if its unipotent radical Ru (G) is trivial (in both the cases, an algebraic group is assumed to be connected). A connected semisimple algebraic group is known (for example, see [25, Theorem 27.5]) to be a central product of to be a connected simple algebraic groups while a connected reductive algebraic group G is known 0 central product of a torus S and a semisimple group M , with S = Z(G) , M = G, G , and S ∩ M finite. If G is a connected reductive algebraic group then let T be its maximal torus (by a torus we always mean a connected diagonalizable (d-) group). The dimension of a maximal torus is called a rank of an algebraic group. The root system of G with respect to a maximal torus T is denoted by Φ(G) (it does not depend on the choice of a maximal torus), and W (G) ≃ NG (T )/T is the Weyl group of G. If G is a reductive group of rank n then the dimension of the centralizer of any its element is not less than n. An element is called regular if the dimension of its centralizer is equal to n. In particular, a semisimple element s is regular if CG (s)0 is a maximal torus of G. Recall that, for every root Psystem Φ, there exists a set of roots r1 , . . . , rn such that each root of Φ can be uniquely written as ni=1 αi ri , where all the coefficients αi are integers and either nonnegative or nonpositive. Such a set of roots is called a fundamental set of Φ, and its elements are called fundamental roots. A fundamental set is a basis of ZΦ ⊗Z R at that. The dimension of ZΦ ⊗Z R is called the rank of Φ. Note that the ranks of G and of its root system Φ(G) are equal. Below we assume that all fundamental roots are positive. Then a root r is positive if and only if it is a linear combination of fundamental roots with nonnegative coefficients. For Pna root system Φ, the set of all positive Pn(negative) + − roots is denoted by Φ (Φ ). The number h(r) = i=1 αi is called the height of r = i=1 αi ri . In every irreducible root system Φ, there exists a unique root of maximal height which is denoted by r0 . Note that the Weyl group W (Φ) of a root system Φ is generated by the reflections in fundamental roots which are called fundamental reflections. If we denote by l(w) the minimal number of multipliers in a decomposition of w into a product of the fundamental reflections, i. e., the length of w, then there exists a unique element of maximal length denoted by w0 , that is a unique element of the Weyl group mapping all positive roots into negative roots. In general, l(w) is equal to |Φ− ∩ (Φ+ )w |, i. e., to the number of positive roots that w maps into negative roots. Let G be a connected simple algebraic group, let π be its exact rational representation, and let Γπ be a lattice generated by the weights of the representation π. By Γad we denote the lattice generated by the roots of Φ, and by Γsc we denote the lattice generated by the fundamental weights. The lattices Γsc , Γπ , and Γad do not depend on the representation of G, and the following inclusions Γad ≤ Γπ ≤ Γsc hold (see [25, 31.1]). It is known that, for a given root system, there are exist several distinct algebraic groups called isogenies. They differs by the structure of Γπ and the order of the finite center. If Γπ coinsides with SIBERIAN ADVANCES IN MATHEMATICS

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Γsc then the group G is said to be simply connected, it is denoted by Gsc . If Γπ coinsides with Γad then the group G is said to have an adjoint type, it is denoted by Gad . Every linear algebraic group with a root system Φ can be obtained as a factor group of Gsc by a subgroup of its center. The center of Gad is trivial and this group is simple as an abstract group. The factor group Γsc /Γπ is denoted by ∆(G) and is called a fundamental group of G. The factor group Γsc /Γad depends on the root system Φ only and is denoted by ∆(Φ). The group ∆(Φ) is known to be cyclic except the root system Φ = D2n when ∆(D2n ) = Z2 × Z2 is elementary Abelian of order 4. Let B be a Borel subgroup, let T ≤ B be a maximal torus, and let U = Ru (B) be a maximal − − connected unipotent subgroup of G. There exists a unique Borel subgroup B such that B ∩ B = T . − − − Denote U = Ru (B ). If we fix an order on Φ(G) then each element u ∈ U (respectively, u ∈ U ) can be uniquely written in the form Y u= xr (tr ) (1) r∈Φ+

Q (respectively, u = r∈Φ− xr (tr )), where the roots are taken in the given order, the elements tr are in the definition field of G, and {Xr , r ∈ Φ} is a set of 1-dimensional T -invariant subgroups (a set of root subgroups). The multiplication of elements from distinct root subgroups is defined by Chevalley commutator formula. Lemma 1.3.1 [5, 5.2.2] (Chevalley commutator formula). Let xr (t) and xs (u) be elements from distinct root subgroups Xr and Xs , respectively, with r 6= −s. Then Y [xr (t), xs (u)] = xir+js (Cijrs (−t)i uj ), ir+js∈Φ;i,j>0

where the constants Cijrs do not depend on t and u. Substantially this formula means that the mutual commutant of Xr and Xs is in the group generated by the subgroups Xir+js , where i, j > 0 and ir + js ∈ Φ. Let ci be the coefficient of a fundamental root ri in the decomposition of r0 . The primes dividing the coefficients ci , are called bad primes. The diagram obtained from the Dynkin diagram by addition of −r0 and its connection with other fundamental roots by usual rule, is called an extended Dynkin diagram. Let R be a (connected) reductive subgroup of maximal rank of a connected simple algebraic group G. As we already noted, in this case, R = G1 ◦ . . . ◦ Gk ◦ Z(R)0 , where Gi -s are connected simple algebraic groups of rank less, than the rank of G. Moreover if Φ1 , . . . , Φk are the respective root systems of G1 , . . . , Gk then Φ1 ∪ . . . ∪ Φk is a subsystem of Φ. There exists a nice algorithm obtained by A. Borel and J. de Siebental [2], and independently by E. B. Dynkin [18], to determine subsystems of a root system. One needs to extend the Dynkin diagram to the extended Dynkin diagram, remove some vertices from it, and repeat the procedure for the connected components obtained. The diagrams obtained in this way are subsystem diagrams, and the diagram of any subsystem can be obtained similarly. In Table 1, we give the extended Dynkin diagrams of all irreducible root systems and the coefficients of the fundamental roots in the decomposition of r0 . The numberring in Table 1 is chosen as that in [13]. For every semisimple element s ∈ G, where G is a connected reductive group, the unit component CG (s)0 is a reductive subgroup of maximal rank and CG (s)/CG (s)0 ≃ D ≤ ∆(G) (see Lemma 1.5.2 below). SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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Table 1. Root systems and extended Dynkin diagrams Extended Dynkin diagram

Φ

−r0 u P PP PP -1 PP PP PP r r3 r1 r2 PPnu u u u ppppppppppppppppppp 1 1 1 1

An

r1 u aa aa r2 1 au ! −r0 !!! 2 ! u -1 r1 −r0 u u i 2 -1 r1 u aa aa r2 1 au ! −r0 !!! 2 ! u -1 r1 r3 u u 2 1

Bn

Cn

Dn

E6

r3 u 2

ppppppppppppppppppp

rn−1 u 2

i

rn u 2

r2 u 2

ppppppppppppppppppp

rn−1 u 2

h

rn u 1

r3 u 2

rn−1 u ! rn−2 !! 1 ! u! a aa 2 aarnu 1 r6 u 1

ppppppppppppppppppp

r5 u 2

r4 u 3 2 u r2 -1 u-r0

E7

E8

F4

G2

-r0 u -1

r1 u 2

r3 u 4 -r0 u -1

r4 u 6 3 u r2 r1 u 2 -r0 u -1

r2 u 3 r1 u 2

r7 u 3

r6 u 4

r5 u 5

i

i

r3 u 4

r7 u 1

r6 u 2

r5 u 3

r4 u 4 2 u r2

r3 u 3

r1 u 2

r8 u 2

-r0 u -1

r4 u 2

r2 u 3

1.4. Structure of finite groups of Lie type . In this Subsection, the notation and definitions for finite groups of Lie type mainly agree with those of [5] (except the definition of finite groups of Lie type). If G is a finite group of Lie type with the trivial center (we do not exclude non-simple groups of Lie type like A1 (2); all the exceptions are given in [5, Theorems 11.1.2 and 14.4.1] and cited in Table 2 b denotes the group of inner-diagonal automorphisms of G. In view of [37, 3.2], Aut(G) below) then G is generated by the inner-diagonal, field, and graph automorphisms. Note that the definition of field and SIBERIAN ADVANCES IN MATHEMATICS

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graph automorphisms in the present paper is slightly different from the definitions given in [37]; precise definitions are given in Subsection 4.1. Since we assume that Z(G) is trivial, G is isomorphic to the b ≤ Aut(G). group of its inner automorphisms and so we may assume that G ≤ G Table 2. Groups of Lie type which are not simple Group Properties A1 (2) the group is solvable A1 (3) the group is solvable B2 (2) B2 (2) ≃ Sym6 G2 (2) [G2 (2), G2 (2)] ≃ 2 A2 (3) 2

A2 (2) the group is solvable

2

B2 (2) the group is solvable

2

G2 (3) [2 G2 (3), 2 G2 (3)] ≃ A1 (8)

2

F4 (2) [2 F4 (2), 2 F4 (2)] is the simple Tits group

Let G be a simple connected algebraic group over an algebraic closure Fp of a finite field of positive characteristic p. Here Z(G) may be nontrivial. A surjective endomorphism σ of G is called a Frobenius ′ map if Gσ is finite. In particular, σ is an automorphism of G as an abstract group. The groups Op (Gσ ) ′ are called canonical finite groups of Lie type, and every group G, with Op (Gσ ) ≤ G ≤ Gσ , is called a finite group of Lie type. If G is a simple algebraic group of adjoint type then we shall say that G is ′ also of adjoint type. Note that, in [5], only the groups Op (G) are called finite groups of Lie type. But later in [8], R. Carter said that every group Gσ is a finite group of Lie type for every connected reductive ′ group G. Moreover, in [7] and [13], without any comments, every group G, with Op (Gσ ) ≤ G ≤ Gσ , is called a finite group of Lie type. Thus, citing the definition of finite groups of Lie type and of finite canonical groups of Lie type, we intend to clarify the situation here. For example, PSL2 (3) is a canonical finite group of Lie type and PGL2 (3) is a finite group of Lie type. Note that an element of order 3 is not conjugate to its inverse in PSL2 (3) and is conjugate to its inverse in PGL2 (3). Since an information about the conjugation is important in many cases (and is very important and useful in the present paper), we find it reasonable to use such notation. In general, given a group of Lie type G (if we consider it as an abstract group), the corresponding algebraic group is not uniquely defined. For example if G = PSL2 (5) ≃ SL2 (4) then G can be obtained ′ either as (SL2 (F2 ))σ or as O5 ((PGL2 (F5 ))σ ) (for suitable σ-s). Hence, for every finite group of Lie type G, we fix (in some way) a corresponding algebraic group G and a Frobenius map σ such that ′ Op (Gσ ) ≤ G ≤ Gσ . We say that the groups 2 An (q), 2 Dn (q), 2 E6 (q) are defined over Fq2 , the groups 3 D4 (q) are defined over Fq3 , and the remaining groups are defined over Fq . The field Fq in all the cases is called a base field. In view of [23, Lemma 2.5.8] if G is of adjoint type then Gσ is the group of inner-diagonal ′ ′ automorphisms of Op (Gσ ). If G is simply connected then Gσ = Op (Gσ ) (see [38, 12.4]). In any case, ′ in view of [23, Theorem 2.2.6(g)], Gσ = T σ Op (Gσ ) for every σ-stable maximal torus T of G. Let U ≤ hXr | r ∈ Φ+ i = U be a maximal unipotent subgroup of G (and U is a maximal connected σstable unipotent subgroup of G at that). Then each u ∈ U can be uniquely written in form (1), where ′ the elements tr are in the definition field of G. If Op (G) coincides with one of the groups 2 An (q), 2 B (22n+1 ), 2 D (q), 3 D (q), 2 E (q), 2 G (32n+1 ), or 2 F (22n+1 ) then we will say that G is twisted, 2 n 4 6 2 4 ′ and in the remaining cases, G is called split. If Op (Gσ ) ≤ G ≤ Gσ is a twisted group of Lie type and r ∈ Φ(G) then by r¯ we always denote the image of a root r under the symmetry of the root system SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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corresponding to the graph automorphism used during the construction of G. Sometimes we will use the notation Φε (q), where ε ∈ {+, −}, and Φ+ (q) = Φ(q) is a split group of Lie type with the base field Fq , and Φ− (q) = 2 Φ(q) is a twisted group of Lie type defined over the field Fq2 (with the base field Fq ). Let R be a connected σ-stable subgroup of G. Then we may consider R = G ∩ R and N (G, R) = G ∩ NG (R). Note that, in general, N (G, R) 6= NG (R), and N (G, R) is called the algebraic normalizer of R. For example, if we consider G = SLn (2) then the subgroup of diagonal matrices H of G is trivial, hence, NG (H) = G. But G = (SLn (F2 ))σ , where σ is a Frobenius map σ : (ai,j ) 7→ (a2i,j ). Then H = H σ , where H is the subgroup of diagonal matrices in SLn (F2 ). Thus N (G, H) is the group of monomial matrices in G. We use the term “algebraic normalizer” in order to avoid such difficulties and to make our proofs to be universal. A group R is called a torus (respectively a reductive subgroup, a parabolic subgroup, and a maximal torus, a reductive subgroup of maximal rank) if R is a torus (respectively a reductive subgroup, a parabolic subgroup, a maximal torus, and a reductive subgroup of maximal rank) of G. A maximal σ-stable torus T of G such that T σ is a Cartan subgroup of Gσ is called a maximal split torus of G. Assume that a reductive subgroup R is σ-stable. In view of [38, 10.10], there exists a σ-stable maximal torus T of R. Let Gi1 , . . . , Giji be a σ-orbit of Gi1 . Consider the induced action of σ on the factor group (Gi1 ◦ . . . ◦ Giji )/Z(Gi1 ◦ . . . ◦ Giji ) ≃ PGi1 × . . . × PGiji . Since PGi1 ≃ . . . ≃ PGiji are simple (as abstract groups) then σ induces a cyclic permutation of the σ set {PGi1 , . . . , PGiji } and we may assume that the numbering is chosen so that PGi1 = PGi2 , . . . , σ PGij = PGi1 . Thus the equality i

j −1

(PGi1 × . . . × PGiji )σ = {x | x = g · gσ · . . . · gσ i

for some g ∈ PGi1 }σ ≃ (PGi1 )σji ′

holds. In view of [38, 10.15], the group PGσji is finite. Hence, Op ((PGi1 )σji ) is a canonical finite group ′ of Lie type, probably, with the base field larger than the base field of Op (Gσ ). Let B i1 be the preimage of a σ ji -stable Borel subgroup of PGi1 in Gi1 under the natural epimorphism, and T i1 be a σ ji -stable maximal torus of Gi1 contained in B i1 (their existence follows from [38, 10.10]). − Then from the note at the beginning of section 11 from [38], the subgroups U i1 and U i1 generated by T i1 invariant root subgroups taken over all positive and negative roots respectively, are also σ ji -stable. Since − Gi1 is a simple algebraic group, Gi1 is generated by the subgroups U i1 and U i1 . Now Z(Gi1 ◦ . . . ◦ Giji ) consists of semisimple elements, so the restriction of the natural epimorphism Gi1 → PGi1 on U i1 and k − − k U i1 is an isomorphism. Therefore, for each k, the subgroups (U i1 )σ and (U i1 )σ are maximal σ ji -stable connected unipotent subgroups of Gik and they generate Gik . −

j −1

−

−

j −1

Thus, U i1 × (U i1 )σ × . . . × (U i1 )σ i and U i1 × (U i1 )σ × . . . × (U i1 )σ i are maximal σ-stable connected unipotent subgroups of Gi1 ◦ . . . ◦ Giji and they generate Gi1 ◦ . . . ◦ Giji . By [38, Corollary 12.3(a)], we have ′

Op ((Gi1 ◦ . . . ◦ Giji )σ ) = j −1

h(U i1 × (U i1 )σ × . . . × (U i1 )σ i

−

−

−

j −1

)σ , (U i1 × (U i1 )σ × . . . × (U i1 )σ i −

)σ i ≃ ′

h(U i1 )σji , (U i1 )σji i = Op ((Gi1 )σji ). −

By [38, 11.6 and Corollary 12.3], the group h(U i1 )σji , (U i1 )σji i is a canonical finite group of Lie − − type. Moreover, from the above arguments the groups h(U i1 )σji , (U i1 )σji i/Z h(U i1 )σji , (U i1 )σji i and ′

′

Op ((PGi1 )σji ) are isomorphic. Denoting Op ((Gi1 ◦ . . . ◦ Giji )σ ) by Gi , we obtain that Gi is a canonical SIBERIAN ADVANCES IN MATHEMATICS

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finite group of Lie type for all i. The subgroups Gi of Op (Gσ ) appearing in this way are called subsystem ′ subgroups of Op (Gσ ). Since Gi1 ◦ . . . ◦ Giji is a σ-stable subgroup, then Gi1 ◦ . . . ◦ Giji ∩ T is a σ-stable maximal torus of Gi1 ◦ . . . ◦ Giji . Therefore we may assume that, for each σ-orbit {Gi1 , . . . , Giji }, the intersection T ∩ Gi1 ◦ . . . ◦ Giji is a maximal σ-stable torus of Gi1 ◦ . . . ◦ Giji . Then Rσ = T σ (G1 ◦ . . . ◦ Gm ) and T σ normalizes each of Gi -s. ′ For a σ-orbit {Gi1 , . . . , Giji } of Gi1 , where Gi = Op ((Gi1 ◦ . . . ◦ Giji )σ ), consider AutRσ (Gi ). Since G1 ◦ . . . ◦ Gi−1 ◦ Gi+1 ◦ . . . ◦ Gk ◦ Z σ ≤ CRσ (Gi ), we have that AutRσ (Gi ) ≃ T σ Gi /Z T σ Gi . From [23, Proposition 2.6.2] it follows that the automorphisms induced by T σ on Gi , are diagonal. di hold. In particular, Aut (Gi ) is a finite group of Therefore, the inclusions PGi ≤ AutRσ (Gi ) ≤ PG Rσ Lie type.

Now, consider the case when L E H ≤ G, where L and H are σ-stable and closed. It is clear that σ induces an action on H/L, and if L is connected then the Lang-Steinberg Theorem (Lemma 1.5.3) implies (H/L)σ = H σ /Lσ . Let R be a σ-stable connected reductive subgroup of maximal rank (in particular, R can be a maximal torus) of G. Since groups NG (R)/R and NW (WR )/WR are isomorphic, where W is the Weyl group of G, and WR is the Weyl group of R (and it is a subgroup of W ), we obtain an induced action of σ on NW (WR )/WR and we say that w1 ≡ w2 for w1 , w2 ∈ NW (WR )/WR if there exists an element w ∈ NW (WR )/WR satisfying the equality w1 = w−1 w2 wσ . Let Cl(Gσ , R) be the set of Gσ g conjugated classes of σ-stable subgroups R , where g ∈ G. Then Cl(Gσ , R) one-to-one corresponds g to the set of σ-conjugate classes Cl(NW (WR )/WR , σ). If w is an element of NW (WR )/WR , and (R )σ g corresponds to the σ-conjugate class of w then (R )σ is said to be obtained by twisting the group R by g the element wσ. Note that and (R )σ ≃ Rσw . The construction of twisting is known and it is given with all necessary results, for example, in [6]. When H = T is a σ-stable maximal torus and W = NG (T )/T then, by [8, Proposition 3.3.6], NG (Tw ) (NG (Tw ))σ ≃ CW,σ (w) = {x ∈ W | σ(x)wx−1 = w}. (2) = Tw (T ) w σ σ Now, assume that the group R is a σ-stable parabolic subgroup of G and U is its unipotent radical. Then it contains a connected reductive subgroup L such that R/U ≃ L. A subgroup L is called a Levi factor of R. Moreover, if Z = Z(L)0 then L = CG (Z) (see [25, 30.2]). Let R(R) be the radical of R. Then it is a σ-stable connected solvable subgroup. Hence, by [38, 10.10], it contains a σ-stable maximal torus Z. Further, CG (Z) = CR (Z) is a σ-stable Levi factor of R. Thus each σ-stable parabolic subgroup of G contains a σ-stable Levi factor L and L is a connected reductive subgroup of maximal rank of G. 1.5. Known results. In this Subsection, we recall some structure results that will be often used below.

Lemma 1.5.1 [26, Theorem 2.2]. Let G be a connected reductive algebraic group, let s ∈ G be a semisimple element of G, and let T be a maximal torus of G containing s. Then CG (s)0 is a reductive subgroup of maximal rank of G. The centralizer G(s) is generated by a torus T , by those T -root subgroups Xr for which sr = e, and by the representatives nw of elements w ∈ W which commute with s. Further, CG (s)0 is generated by the torus T , by those T -root subgroups Xr for which sr = e, and each unipotent element centralizing s, is in CG (s)0 . Lemma 1.5.2 [26, Proposition 2.10]. Let G be a simple algebraic group and s be its semisimple element of finite order. Then the factor group CG (s)/CG (s)0 is isomorphic to a subgroup of the fundamental group ∆(G). In particular, if G is simply connected then CG (s) is connected. Lemma 1.5.3 [38, Theorem 10.1] . Let G be a connected algebraic group and σ be a Frobenius map. Then the map x 7→ x−1 xσ is surjective. SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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The following lemma is known as the Borel–Tits theorem. Lemma 1.5.4. Let X be a subgroup of a finite group of Lie type G such that Op (X) is nontrivial. Then there exists a σ-stable parabolic subgroup P of G such that X ≤ P and Op (X) ≤ Ru (P ). Proof. Define U0 = Op (X) and N0 = NG (U0 ). Then Ui = U · Ru (Ni−1 ) and Ni = NG (Ui ). Clearly, Ui and Ni are σ-stable for all i. In view of [25, Proposition 30.3], the chain of subgroups N0 ≤ N1 ≤ . . . ≤ Nk ≤ . . . is finite and P = ∪i Ni is a proper parabolic subgroup. Clearly, P is σ-stable. Lemma 1.5.5 (the Hartley–Shute lemma [24, Lemma 2.2]). Let G be a finite canonical adjoint group of Lie type with the definition field Fq . Let H be a Cartan subgroup of G and s ∈ Fq . If G is twisted and r = r¯ then assume also that s is in the base field of G. Then there exists an element h(χ) ∈ H such that χ(r) = s except the following cases when h(χ) can be chosen only as follows: (a) G = A1 (q), χ(r) = s2 ; (b) G = Cn (q), r is a long root, χ(r) = s2 ; (c) G = 2 A2 (q), r 6= r¯, χ(r) = s3 ; (d) G = 2 A3 (q), r 6= r¯, χ(r) = s2 ; (e) G = 2 Dn (q), r 6= r¯, χ(r) = s2 ; (f) G = 2 G2 (32n+1 ), r = a or r = 3a + b, where a and b are a short and a long fundamental root respectively, and χ(r) = s2 . Theorem 1.5.6 [33, Theorem 1.1]. Let q = pα , where p is a prime. Assume that G = Spn (q) or SOεn (q) ≤ G ≤ GOεn (q), where q is odd or SUn (q) ≤ G ≤ GUn (q). If G contains a Carter subgroup K then either K is the normalizer of a Sylow 2-subgroup of G or one of the following assertions holds: (a) G ∈ {Sp2 (3), SL2 (3), 2.SU2 (3)} and K is the normalizer of a Sylow 3-subgroup of G; (b) G = GU3 (2) has order 23 · 34 , and K has order 2 · 32 . Moreover, if G is orthogonal then K is a 2-group except possibly the case when G = SOε2 (q). 2. CONJUGACY CRITERION FOR CARTER SUBGROUPS 2.1. Main results. Definition 2.1.1. A finite group G is said to satisfy condition (C) if, for every non-Abelian composition factor S of every composition series of G and for every nilpotent subgroup N of G, the Carter subgroups of hAutN (S), Si are conjugate (in particular, they may not exist). Lemma 2.1.2. Let H be a normal subgroup of a finite group G, let B ⊳ A ≤ G, let S = (A/H)/(B/H) be a composition factor of G/H, and finally, let L ≤ G. Then AutL (A/B) ≃ AutLH/H ((A/H)/(B/H)). Proof. Since H ≤ B then H ≤ CG (A/B), so we may assume that L = LH. Further, we may assume that L ≤ NG (A) ∩ NG (B) and G = LA. Then the action on A/B given by x : Ba 7→ Bx−1 ax coincides with the action on (A/H)/(B/H) given by xH : BaH 7→ Bx−1 axH, and the assertion is proven. The following lemma is known. Lemma 2.1.3. Let G be a finite group, H be a normal subgroup of G and N be a nilpotent subgroup of G = G/H. Then there exists a nilpotent subgroup N of G such that N H/H = N . Proof. Clearly, we may assume that G/H = N . There exists a subgroup U of G such that U H = G. Choose a subgroup of minimal order with this property. Then U ∩ H is contained in the Frattini subgroup F of U . Indeed, if there exists a maximal subgroup M of U not containing U ∩ H then M H = G that contradicts the minimality of U . Thus the group U/F is nilpotent, and hence, U is nilpotent and N = U . SIBERIAN ADVANCES IN MATHEMATICS

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From Lemmas 2.1.2 and 2.1.3 it follows that if a finite group G satisfies (C) then, for every its normal subgroup N and solvable subgroup H, the groups G/N and HN satisfy (C). In this Section, we prove that if G satisfies (C) then all its Carter subgroups are conjugate. More precisely, the following theorem will be proven. Theorem 2.1.4. If a finite group G satisfies (C) then the Carter subgroups of G are conjugate. In Subsections 2.2 and 2.3 below, we assume that X is a counterexample of minimal order to Theorem 2.1.4, i. e., that X is a finite group satisfying (C), and X contains nonconjugate Carter subgroups but the Carter subgroups in every group M of order less than |X| satisfying (C) are conjugate. 2.2. Preliminary results. Lemma 2.2.1. Let G be a finite group satisfying (C), |G| 6 |X|, and H be a Carter subgroup of G. If N is a normal subgroup of G then HN/N is a Carter subgroup of G/N . Proof. Since HN/N is nilpotent, we have just to prove that it is self-normalizing in G/N . Clearly, this is true if G = HN . So, assume M = HN < G (note that, by Lemmas 2.1.2 and 2.1.3, the group M satisfies (C)). By the minimality of X, the equality M x = M , with x ∈ G, implies that H x = H m for some m ∈ M . From here it follows that xm−1 ∈ NG (H) = H and x ∈ M . This proves that HN/N is nilpotent and self-normalizing in G/N . Lemma 2.2.2. Let B be a minimal normal subgroup of X, and let H and K be non-conjugate Carter subgroups of X. Then (1) B is insoluble; (2) X = BH = BK; (3) B is the unique minimal normal subgroup of X. Proof. (1) We give a proof by contradiction. Assume that B is soluble and let π : X → X/B be the canonical homomorphism. Then H π and K π are Carter subgroups of X/B by Lemma 2.2.1. By the minimality of X, there exists x ¯ = Bx such that (K π )x¯ = H π . It implies K x ≤ BH. Since BH is soluble, x K is conjugate to H in BH, hence, K is conjugate to H in X, a contradiction. (2) Assume that BH < X. By Lemma 2.2.1 and the minimality of X, BH/B and BK/B are conjugate in X/B: So, there exists x ∈ X such that K x ≤ BH. From here it follows that K x is conjugate to H in BH. Hence, K is conjugate to H in X, a contradiction. (3) Suppose that M is a minimal normal subgroup of X different from B. By (1), M is insoluble. On the other hand, M B/B ≃ M is a subgroup of the nilpotent group X/B ≃ H/H ∩ B, a contradiction. Lemma 2.2.3. Let K be a Carter subgroup of a finite group G. Assume that there exists a normal subgroup B = T1 × . . . × Tk of G such that G = KB, Z(Ti ) = {e}, and Ti is not decomposable into a direct product of its proper subgroups for all i. Then AutK (Ti ) is a Carter subgroup of hAutK (Ti ), Ti i. Proof. Assume that our statement does not hold and G is a counterexample, with k minimal. Then k > 1. Since each group Ti has a trivial center and is not decomposable into a direct product of proper subgroups, a corollary of the Krull–Remak–Shmidt theorem [34, 3.3.10] implies that the action by conjugation of G on the set {T1 , . . . , Tk } induces permutations of this set. Clearly, by conjugation, G acts transitively on the set Ω := {T1 , . . . , Tk }. We may assume that the Tj -s are indexed so that G acts primitively on the set {∆1 , . . . , ∆p }, p > 1, where, for each i, ∆i := {T1+(i−1)l , . . . , Til },

k = pl.

Denote by ϕ : G → Symp the induced permutation representation. Clearly, B ≤ ker ϕ, so that Gϕ = (BK)ϕ = K ϕ is a primitive nilpotent subgroup of Symp . Hence, p is prime and Gϕ is a cyclic group of order p. In particular, Y := ker ϕ coincides with the stabilizer of any ∆i , so that ϕ is permutationally equivalent to the representation of G on the right cosets of Y . For each i = 1, . . . , p, let Si = T1+(i−1)l × . . . × Til . Then Y = NG (Si ) and B = S1 × . . . × Sp . Consider ξ : Y → AutY (S1 ) and set A = Y ξ and S = S1ξ . Clearly, S is a normal subgroup of A; moreover, S is isomorphic to S1 since S1 has a trivial center. On the other hand, for each i 6= 1, we have Si ≤ ker ξ since Si centralizes S1 . SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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Denote by A ≀ Zp the wreath product of A and a cyclic group Zp , and let {x1 = e, . . . , xp } be a right transversal of Y . Then the map η : G → A ≀ Zp such that, for each x ∈ G, ξ ξ −1 x1 xx−1 , . . . , x xx η : x 7→ xϕ ϕ ϕ p 1x px is a homomorphism. Clearly, Y η is a subdirect product of the base subgroup Ap and S1η = {(s, 1, . . . , 1) | s ∈ S}, B η = {(s1 , . . . , sp ) | si ∈ S} ≤ Y η . Moreover, ker η = CG (B) = {e}, so we may identify G with Gη . We choose h ∈ K \ Y . Then G = hY, hi, hp ∈ Y, K = (Y ∩ K)hhi and we may assume that h = (a1 , a2 , . . . , ap )π, ai ∈ A, π = (1, 2, . . . , p) ∈ Zp . For each i, 1 6 i 6 p, let ψi : Ap → A be the canonical projection and let Ki := (K ∩ i−1 a ...a Y )ψi . Clearly, Y ψi = A. Moreover, for each i > 2, Ki = K1h = K1 1 i−1 since h normalizes Y ∩ K. Let N := (K1 × . . . × Kp ) ∩ Y. Since K = (N ∩ K)hhi and Kih = Ki+1 (mod p) , the subgroup N is normalized by K. We claim that K1 is a Carter subgroup of A. Assume that n1 ∈ NA (K1 ) \ K1 . From Y = (Y ∩ K)B it follows that n1 = h1 s, h1 ∈ K1 , and s ∈ NS (K1 ) \ K1 . Let b := (s, sa1 , . . . , sa1 ...ap−1 ) ∈ B. Then b normalizes N since Kib = Kis

a1 ...ai−1

a ...ai−1 sa1 ...ai−1

= K1 1

sa1 ...ai−1

= K1

a ...ai−1

= K1 1

= Ki .

Further, [b, h−1 ] := b−1 hbh−1 ∈ Y is such that [b, h−1 ]ψi = 1 if i 6= p, [b, h−1 ]ψp = [s, (a1 · . . . · ap )−1 ]a1 ·...·ap−1 , where a1 · . . . · ap = (hp )ψ1 ∈ K1 . Since s ∈ NS (K1 ), we have [s, (a1 · . . . · ap )−1 ] ∈ K1 , [s, (a1 · . . . · ap )−1 ]a1 ·...·ap−1 ∈ Kp . So, [b, h−1 ] ∈ N and b ∈ NG (N hhi). But the inclusion K ≤ N hhi implies NG (N hhi) = N hhi. Indeed, if g ∈ NG (N hhi) then K g is a Carter subgroup of N hhi. But N hhi is soluble. Hence there exists y ∈ N hhi, with K g = K y . Further, K is a Carter subgroup of G, thus gy −1 ∈ K and g ∈ N hhi. Therefore, b ∈ N, s ∈ K1 , i. e., n1 ∈ K1 which is a contradiction. Now, let A = K1 (T1 × . . . × Tl ) and l < k. We can prove by induction that AutK1 (T1 ) is a Carter subgroup of hAutK1 (T1 ), T1 i. In view of our construction, AutK (T1 ) = AutK1 (T1 ) and the proof is complete. 2.3. Proof of Theorem 2.1.4 . Recall that B = T1 × · · · × Tk , where Ti ≃ T is a non-Abelian simple group. What remains to prove is that k = 1. In the notation of the proof of Lemma 2.2.3, we have shown that H1 is a Carter subgroup of A. If k > 1 then |A| < |X| and A satisfies (C). So each Ki is conjugate with K1 in A and NA (Ki ) = Ki , i = 1, . . . , p. From here it follows easily that N is a Carter subgroup of Y . Let y := (y1 , . . . , yp ) ∈ NY (N ). From the equality N ψi = Ki we have yi ∈ NA (Ki ) = Ki for each i. Hence, y ∈ N . We proved that, for each Carter subgroup K of X, there is a Carter subgroup N = NK of Y such that K normalizes NK . Clearly, NK 6= {e}, otherwise X would be of order p. So, let H be a Carter subgroup of X not conjugate to K, and let NH be a Carter subgroup of Y corresponding to H. If k > 1 then Y is a proper subgroup of X and Y satisfies (C). By the minimality of X, we obtain that NH and NK are conjugate in Y and we may assume that NH = NK . Then HNH = KNH is solvable. Hence the subgroups H and K are conjugate. This contradiction completes the proof of Theorem 2.1.4. 2.4. Some properties of Carter subgroups . Here we will prove some lemmas that will be useful in studying Carter subgroups in finite groups, in particular, in almost simple groups. Lemma 2.4.1. Let K be a Carter and N be a normal subgroups of a finite group G. Assume that KN satisfies (C) (this condition holds if either G satisfies (C) or N is solvable) or KN = G. Then KN/N is a Carter subgroup of G/N . SIBERIAN ADVANCES IN MATHEMATICS

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Proof. If KN = G then the statement is evident. Assume that KN 6= G, i. e., KN satisfies (C). Consider x ∈ G and assume that xN ≤ NG/N (KN/N ). Therefore, x ∈ NG (KN ). We have that K x is a Carter subgroup of KN . Since KN satisfies (C), we obtain that its Carter subgroups are conjugate. Thus there exists y ∈ KN such that K y = K x . Since K is a Carter subgroup of G, it follows from here that xy −1 ∈ NG (K) = K and x ∈ KN . Lemma 2.4.2. Let K be a Carter subgroup of a finite group G. Assume also that e 6= z ∈ Z(K) and CG (z) satisfies (C). Then (1) Every subgroup Y which contains K and satisfies (C), is self-normalizing in G; (2) None of the elements conjugate to z in G except z, lies in Z(G); (3) If H is a Carter subgroup of G nonconjugate to K then z is not conjugate to any element from the center of H. In particular, the centralizer CG (z) is self-normalizing in G, and z is not conjugate to any power z k 6= z. Proof. (1) Take x ∈ NG (Y ). Then K x is a Carter subgroup of Y . By Theorem 2.1.4, the Carter subgroups of Y are conjugate. Therefore there exists y ∈ Y with the property K x = K y . Hence, xy −1 ∈ NG (K) = K ≤ Y and x ∈ Y. −1

(2) Assume that z x ∈ Z(K) for some x ∈ G. Then z belongs to the center of hG, Gx i ≤ CG (z). Since CG (z) satisfies (C), there exists y ∈ CG (z) such that K x = K y . From the fact xy −1 ∈ CG (z) we −1 −1 obtain that z xy = z. Hence, z x = z y = z. So, we conclude that z x = z. (3) If our claim is not valid then replacing H with some conjugate H x (if necessary), we may assume z ∈ Z(K) ∩ Z(H), i. e., z ∈ Z(hK, Hi) ≤ CG (z). Since CG (z) satisfies (C), there exists y ∈ CG (z) such that H = K y , a contradiction. Note that, for every known finite simple group G (and hence, for almost simple ones since the group of outer automorphisms is soluble) and, for all elements z ∈ G of prime order, we see that the composition factors of CG (z) are known simple groups. Indeed, for sporadic groups this statement can be checked by using [10]. The composition factors of CAn (z) are alternating groups. If G is a finite simple group of Lie type over a field of characteristic p and (|z|, p) = 1 then z is semisimple and all the composition factors of CG (z) are finite groups of Lie type. If |z| = p and p is a good prime for G then, by Theorems 1.2 and 1.4 from [36], all the composition factors of CG (z) are finite groups of Lie type. From papers of several authors it follows that, in the case when p is a bad prime for a finite adjoint group of Lie type G, all the composition factors of the centralizer of an element of order p are known finite simple groups. Therefore, if we are classifying Carter subgroups of an almost simple group A then we may assume that CA (z) satisfies (C) for all elements z ∈ A of prime order. Lemma 2.4.3. Let Q be a Sylow 2-subgroup of a finite group G. Then G contains a Carter subgroup K, with Q ≤ K, if and only if NG (Q) = QCG (Q). Proof. Assume that G contains a Carter subgroup K satisfying Q ≤ K. Since K is nilpotent, it follows that Q is normal in K and K ≤ QCG (Q) E NG (Q). By the Feit-Thompson theorem (see [20]), we obtain that NG (Q) is solvable. Thus, by Lemma 2.4.2 (1), we have that QCG (Q) is self-normalizing in NG (Q), so, NG (Q) = QCG (Q). Assume now that NG (Q) = QCG (Q), i. e., the equality NG (Q) = Q × O(CG (Q)) holds. Since O(CG (S)) is of odd order, it is solvable. Therefore it contains a Carter subgroup K1 . Consider the nilpotent subgroup K = Q × K1 of G. Assume that x ∈ NG (K) then x ∈ NG (Q). But K is a Carter subgroup of NG (Q), hence, x ∈ K and K is a Carter subgroup of G. Definition 2.4.4. A finite group G is said to satisfy (ESyl2) if for its Sylow 2-subgroup Q the equality NG (Q) = QCG (Q) holds. In other words, G satisfies (ESyl2) if every element of odd order normalizing a Sylow 2-subgroup Q of G, centralizes Q. Lemma 2.4.5. Let Q be a Sylow 2-subgroup of a finite group G and x be an element of odd order from NG (Q). Assume that there exist normal subgroups G1 , . . . , Gk of G such that G1 ∩ . . . ∩ Gk ∩ Q ≤ Z(NG (Q)) and x centralizes Q modulo Gi for all i. Then x centralizes Q. In particular, if G/Gi satisfies (ESyl2) for all i then G satisfies (ESyl2). SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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Proof. Consider the normal series Q D Q1 D . . . D Qk D Qk+1 = {e}, where Qi = Q ∩ (G1 ∩ . . . ∩ Gi ). The conditions of the lemma imply that x centralizes each factor Qi−1 /Qi . Since x is an element of odd order, this implies that x centralizes Q. Lemma 2.4.6. Let H be a subgroup of a finite group G such that |G : H| = 2t , H satisfies (ESyl2), and each element of odd order of G is in H (this condition is evidently equivalent to the subnormality of H). Then G satisfies (ESyl2). Proof. Let Q be a Sylow 2-subgroup of G such that Q ∩ H is a Sylow 2-subgroup of H. Consider an element x ∈ NG (Q) of odd order. Since x ∈ H then x ∈ NH (Q) ≤ NH (Q ∩ H) = (Q ∩ H) × O(NH (Q ∩ H)), i. e., x ∈ O(NH (Q ∩ H)). Thus the set of elements of odd order in NG (Q) forms a subgroup R = O(NH (Q ∩ H)) ∩ NG (Q) of NG (Q). Clearly, R is normal in NG (Q), Therefore, R = O(NG (Q)). On the other hand, Q is normal in NG (Q) by definition and Q ∩ R = {e} whence NG (Q) = Q × O(NG (Q)). Using the results of this Section, we will improve the definition of a minimal counter example. Definition 2.4.7. A finite almost simple group A is called a minimal counter example if it contains nonconjugate Carter subgroups but the Carter subgroups of every almost simple group of order less than |A|, with the simple socle being a known simple group, are conjugate. 3. CONJUGACY IN SIMPLE GROUPS 3.1. Brief review of the results . Recall that, in view of Lemma 2.4.2, none of the elements from the center of a Carter subgroup can be conjugate to its nontrivial power (if the centralizer of the element satisfies (C)). Thus if we would be able to prove that each element of prime order r of G is conjugate to its nontrivial power and at the same time its centralizer satisfies (C) then we may state that the order of a Carter subgroup (if it exists) is not divisible by r. In this Section, we obtain an information concerning the conjugacy of elements of prime order in finite simple groups, and using this information, we obtain a description of Carter subgroups in a broad class of almost simple groups. Actually, in almost simple groups distinct from Aεn (q) (ε = ±), Carter subgroups must be 2-groups as will be clear below. The results can be formulated as a list of almost simple groups A which cannot be minimal counter example (see Theorem 3.3.5). This list is summarized in Table 3, where Field(S) denotes the group generated by the field and inner-diagonal automorphisms of a finite group of Lie type S. Table 3. Finite almost simple groups which are not minimal counter examples Soc(A)= G

Conditions for A

alternating, sporadic; A1 (pt ), Bℓ (pt ), Cℓ (pt ), t is even if p = 3; 2

B2 (22n+1 ),G2 (pt ), F4 (pt ), 2 F4 (22n+1 ); E7 (pt ), p 6= 3; E8 (pt ), p 6= 3, 5 3

none

D4 (pt ), D2ℓ (pt ), 2 D2ℓ (pt ),

t is even if p = 3 in the last 2 cases and, b ∩ A)|2′ > 1 if G = D4 (pt ), |(Field(G) ∩ A) : (G Bℓ (3t ), Cℓ (3t ), D2ℓ (3t ), 3 D4 (3t ), 2 D2ℓ (3t ), D2ℓ+1 (rt ), 2 D2ℓ+1 (rt ), 2 G2 (32n+1 ),

A=G

E6 (pt ), 2 E6 (pt ), E7 (3t ), E8 (3t ), E8 (5t )

In particular, A cannot be simple (the case A = Aεℓ (q) is excluded by Theorem 1.5.6). SIBERIAN ADVANCES IN MATHEMATICS

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3.2. Preliminary results. Lemma 3.2.1. Let G be a simple connected algebraic group over a field of characteristic p and let t be an element of order r of G not divisible by p. Then CG (t)/CG (t)0 is a π(r)-group. Proof. Since p does not divide r then t is semisimple. By Lemma 1.5.1, CG (t)0 is a connected reductive subgroup of maximal rank of G and every p-element of CG (t) is contained in CG (t)0 . Assume that a prime s 6∈ π(r) divides |CG (t)/(CG (t)0 )|. Then s 6= p and CG (t) contains an element x of order sk such that x 6∈ CG (t)0 . Since x and t commute, we have that x · t is a semisimple element of G (of order r · sk ). Therefore, there exists a maximal torus T of G containing x · t. Then (xt)r = xr ∈ T . Since (s, r) = 1, there exists m such that rm ≡ 1 (mod sk ). Thus, (xr )m = x ∈ T . Since xt, x ∈ T then t ∈ T , so T ≤ CG (t)0 . Hence, x ∈ CG (t)0 , a contradiction. The following statement is a direct consequence of Lemmas 1.5.2 and 3.2.1. Lemma 3.2.2. Let s ∈ G be a semisimple element of order r such that (r, ∆(G)) = 1. Then CG (s) is connected. In particular, from here it follows that, for every Frobenius map σ of G, two semisimple elements s and s′ ∈ Gσ are conjugate in Gσ if and only if they are conjugate in G. The following lemma plays an important role since it shows that a semisimple element of odd prime order is usually conjugate to its inverse. ′

Lemma 3.2.3. Let G = Or (Gσ ), let G have an adjoint type, and let the root system of G have b a type distinct from Aℓ (ℓ > 1), D2ℓ+1 , and E6 . Then each semisimple element of odd order s ∈ G is conjugate to its inverse by an element of G.

Proof. There exists some σ-stable maximal torus T of G, with s ∈ T , such that T is generated by the ∗ set {hα (λ) | α ∈ Φ, λ ∈ Fp } and the factor group NG (T )/T is isomorphic to the Weyl group W of G. If w ∈ W and nw is a preimage of w under the natural epimorphism NG (T ) → W then hα (λ)nw = hαw (λ). Now, let w0 be the unique involution of W such that w0 (Φ+ ) = Φ− and let n0 be a preimage of w0 . Since we are assuming Φ 6= Aℓ (ℓ > 1), D2ℓ+1 , and E6 , we have αw0 = −α for all α ∈ Φ. Hence, hα (λ)n0 = h−α (λ) = hα (λ)−1 . We conclude that sn0 = s−1 , i.e., that s is conjugate to s−1 in G. Thus, by the previous lemma, s and s−1 are conjugate in Gσ . Finally, from Gσ = T σ G we conclude that s and s−1 are conjugate in G. Lemma 3.2.4. Let C be a connected reductive subgroup of maximal rank of G. Denote by W and WC the Weyl groups of G and C respectively, by WC⊥ the subgroup of W generated by the reflections in roots orthogonal to all roots from Φ(C), and by ∆C the Dynkin diagram of C. Then (a) NW (WC )/(WC × WC⊥ ) ≃ AutW (∆C ); (b)

NG (C)/C ≃ NW (WC )/WC . ′

Let G = Op (Gσ ) be split or coincide with one of the following groups: 2 Aℓ (pt ), 2 D2ℓ+1 (pt ), and 2 E (pt ). If s ∈ G is a semisimple element such that C (s) is connected and N (C (s)) > C (s) 6 G G G G then NG (CG (s)) > CG (s). Proof. Item (a) can be found in [6, Proposition 4]. As to item (b), let T be a maximal torus of G contained in C, so that we may assume that W = NG (T )/T and WC = NC (T )/T . All maximal tori of C are conjugate in C since C is connected. It follows easily that NG (C) = CNN (T ) (C). Moreover, it is G

shown in [6, Proposition 5] that NN

(C) G (T )

= NN

(NC (T )). G (T )

Hence,

NN CNN (T ) (NC (T )) NG (C) NW (WC ) G(T ) (NC (T )) G = ≃ ≃ . WC C C NC (T ) ′

(3)

Now, let G = Op (Gσ ) be as in the statement, and set C = CG (s). Write σ = τ ϕ, where τ is the graph automorphism of G induced by a symmetry ρ of the Dynkin diagram of Φ = Φ(G) and ϕ is a SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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field automorphism. Now, let τ be the isometry which extends ρ on Euclidean space R ⊗Z ZΦ. If T 1 is a σ-stable maximal split torus of G then, for each x ∈ NG (T 1 )/T 1 , we have xσ = τ x (considering NG (T 1 )/T 1 = W1 as a group of isometries of R ⊗Z ZΦ). Thus if G is split, i. e., ρ = τ = e, then σ acts trivially on W1 . If G is twisted, hence, is of type Aℓ , D2ℓ+1 , or E6 , it is possible to show directly that −τ ∈ W1 . Thus we may twist T 1 by −τ obtaining the σ-stable torus (T 1 )−τ . By equation (2), (NG ((T 1 )−τ )))σ ≃ CW1 ,σ (−τ ) = {x ∈ W1 | τ x(−τ )x−1 = −τ } = W1 . ((T 1 )−τ ))σ Let {Xα | α ∈ Φ} be the set of T 1 -root subgroups and set C 1 = h T 1 , Xα | α ∈ Φ(C) i. Since Φ(C) is σ-invariant, it follows that C 1 is σ-stable. Moreover, since τ (Φ(C)) = Φ(C), we have that −τ ∈ NW1 (WC 1 ). By [6, Proposition 1 and 2], it follows that there exists (C 1 )−τ obtained from C 1 by twisting with −τ . Up to conjugation in G, we can assume that (T 1 )−τ ≤ (C 1 )−τ . Define T 0 = T 1 and C 0 = C 1 if G is split, and T 0 = (T 1 )−τ and C 0 = (C 1 )−τ if G is twisted. Since Φ(C) = Φ(C 0 ), there exists g ∈ G, such that g C 0 = C and g T 0 = T . It follows that w˙ = g−1 σ(g) ∈ NG (C 0 ) ∩ NG (T 0 ). So the image w of w˙ in W0 = NG (T 0 )/T 0 belongs to NW0 (WC0 ). From Gσ = T σ G it follows that (NG (C))σ = NGσ (C) = T σ NG (C). Hence we will complete the proof if we can show that the group

(NG (C))σ Cσ

=

is nontrivial. Using equation (3), we get

T σ NG (C) T σ CG (s)

≃

NG (C) CG (s)

which is a subgroup of

NG (CG (s)) CG (s) ,

(NN (T ) (NC (T ))/T )σ NN (T ) (NC (T ))/T ∩ (NG (T )/T )σ (NG (C))σ G G ≃ ≃ . Cσ (NC (T )/T )σ NC (T )/T ∩ (NG (T )/T )σ NG (T 0 ) N (T ) , i. e., σ acts trivially on the finite group By our choice of T 0 , we have GT 0 = T 0

0

σ

NG (T 0 ) . T0

Now, if w ∈ WC 0 then, by [6, Proposition 1], one can assume that w = e, T = T 0 , C = C 0 . From here it follows NG (T )/T = (NG (T )/T )σ . Hence, NW (WC ) N (C) (NG (C))σ ≃ ≃ G WC Cσ C which is non-trivial by assumption. Finally, assume that w 6∈ WC 0 , i. e. w˙ = g−1 σ(g) 6∈ C 0 . It follows that g w˙ = σ(g)g−1 6∈ C, i. e., g wT ˙ 6∈ NC (T )/T . On the other hand g w˙ ∈ NG (C) ∩ NG (T ). Moreover, ˙ 0 ) = wT ˙ 0 , i. e., σ(g)−1 gσ(g)−1 σ 2 (g) = t0 ∈ T 0 . since σ acts trivially on NG (T 0 )/T 0 , we have that σ(wT −1 Hence, g t0 = t ∈ T and σ(g)g t = (g w) ˙ −1 σ(g w) ˙ ∈ T . It follows that σ(g wT ˙ ) = g wT ˙ . So if w 6∈ WC 0 , we g conclude that wT ˙ maps onto a non trivial element of the group NN

G (T )

(NC (T ))/T ∩ (NG (T )/T )σ

NC (T )/T ∩ (NG (T )/T )σ

.

The rest of this Subsection is devoted to unipotent elements in groups of Lie type. ′

Lemma 3.2.5. Let G = Op (Gσ ) be a finite group of Lie type with the base field Fpt , with p odd. If p = 3, suppose t to be even. Assume further that Φ(G) 6= G2 , F4 , E6 , E7 , E8 if p = 3, and Φ(G) 6= E8 if p = 5. Then every unipotent element u of order p is conjugate in G to some power uk 6= u. Proof. Under our assumptions, p is a good prime. By item (i) of [36, Theorem 1.4], there exists a ′ closed σ-stable subgroup A1 (Fp ) of G such that u ∈ A1 (Fp ). Clearly, Op ((A1 (Fp ))σ ) is isomorphic either to SL2 (ptm ) or to PSL2 (ptm ) for some positive integer m > 0. Up to conjugation in A1 (Fp ), we SIBERIAN ADVANCES IN MATHEMATICS

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



1 ζ (or u is equal to the projective image of this matrix) for some ζ ∈ Fptm . 0 1   −1 η 0  Under our assumptions, there exists η ∈ Fpt such that 1 6= η 2 = k ∈ Fp . Let x be the matrix  0 η   1 kζ  or its projective image. Then x ∈ G, and u and ux =  = uk are conjugate in G. 0 1 can assume that u = 

Lemma 3.2.6. Let u ∈ G = G2 (3t ) be an element of order 3. Then u is conjugate to u−1 in G. Proof. By [19, Proposition 6.4], there exist 9 unipotent conjugacy classes in G. All of them may be found in Table 4 where α and β denote respectively short and long fundamental roots of G2 , ζ is an element of F3t such that the polynomial x3 − x + ζ is irreducible in F3t [x] and η is a non-square element of F3t . Since |x1 | = 9, and x2 and x3 are conjugate to x1 in G2 (F3 ), we only need to verify 2(α,β)

that x4 , x5 , x6 , x7 , x8 are conjugate to their inverses. Using the formula xβ (u)hα (t) = xβ (t (α,α) u) for h (−1)

each α, β ∈ Φ (see [5, Proposition 6.4.1]), we obtain x6 α

h (−1) x5 β

h (−1)

β = x−1 6 , x8

h (−1)

β = x−1 8 , x4

= x−1 4 , and

= x−1 6 |CG (xi )| for all i 6= 7. Thus, x7 is also conjugate to its inverse. 5 . Finally, |CG (x7 )| = Table 4. Unipotent classes in G2 (q), q = 3t . representative x

|CG (x)|

x0 = 1

q 6 (q 2 − 1)(q 6 − 1)

x1 = xα (1)xβ (1)

3q 2

x2 = xα (1)xβ (1)x3α+β (ζ)

3q 2

x3 = xα (1)xβ (1)x3α+β (−ζ) 3q 2 x4 = xα+β (1)x3α+β (1)

2q 4

x5 = xα+β (1)x3α+β (η)

2q 4

x6 = x2α+β (1)

q 6 (q 2 − 1)

x7 = x2α+β (1)x3α+2β (1)

q6

x8 = x3α+2β (1)

q 6 (q 2 − 1)

Lemma 3.2.7. Let u ∈ G = F4 (3t ) be an element of order 3. Then u is conjugate to u−1 in G. Proof. By [35, Table 6], there exist 28 unipotent conjugacy classes of G. All of them can be found in Table 5. Recall that, in an Euclidean 4-dimensional space with orthonormal base ε1 , ε2 , ε3 , ε4 , all the roots of F4 may be written as {±εi ± εj , ±εi , 21 (±ε1 ± ε2 ± ε3 ± ε4 )}. In Table 5, the symbols ±i ± j, ±i, and ±1 ± 2 ± 3 ± 4 denote the roots ±εi ± εj , ±εi , and 12 (±ε1 ± ε2 ± ε3 ± ε4 ) respectively, η is a fixed non-square element of F3t , ξ is a fixed element of F3t such that x2 + ξx + η is an irreducible polynomial in F3t [x], and ζ is a fixed element of F3t such that x3 − x + ζ is an irreducible polynomial in F3t [x]. Using [35, Table 7], one can easily verify that |x9 | = |x10 | > 3, |xi | > 3 for all i ≥ 12. Indeed, by [35, Table 7], we have that the elements x9 and x10 are conjugate in F4 (F3 ). They also are conjugate to the element c7 = xr1 (1)xr2 (1)xr3 (1), where the roots r1 , r2 and r3 are fundamental roots in a root system of type A3 . But it is evident that |c7 | > 3. In all the cases when |xi | > 3, we act in a similar way. In the remaining cases, one can see that |CG (xi )| = 6 |CG (xj )| for all i 6= j. So, if |xi | = 3 then i = 1, . . . , 8 or i = 11, and xi is conjugate to its inverse in G. SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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Table 5. Unipotent classes F4 (q), q = 3t representative x

|CG (x)|

x0 = 1

|G|

x1 = x1+2 (1)

q 24 (q 2 − 1)(q 4 − 1)(q 6 − 1)

x2 = x1−2 (1)x1+2 (−1)

2q 21 (q 2 − 1)(q 3 − 1)(q 4 − 1)

x3 = x1−2 (1)x1+2 (−η)

2q 21 (q 2 − 1)(q 3 + 1)(q 4 − 1)

x4 = x2 (1)x3+4 (1)

q 20 (q 2 − 1)2

x5 = x2−3 (1)x4 (1)x2+3 (1)

2q 17 (q 2 − 1)(q 3 − 1)

x6 = x2−3 (1)x4 (1)x2+3 (η)

2q 17 (q 2 − 1)(q 3 + 1)

x7 = x2 (1)x1−2+3+4 (1)

q 14 (q 2 − 1)(q 6 − 1)

x8 = x2−3 (1)x4 (1)x1−2 (1)

q 16 (q 2 − 1)

x9 = x2−3 (1)x3−4 (1)x3+4 (−1)

2q 12 (q 2 − 1)2

x10 = x2−3 (1)x3−4 (1)x3+4 (−η)

2q 12 (q 4 − 1)

x11 = x2+3 (1)x1+2−3−4 (1)x1−2+3+4 (1)

q 14 (q 2 − 1)

x12 = x2−3 (1)x4 (1)x1−4 (1)

2q 12 (q 2 − 1)

x13 = x2−3 (1)x4 (1)x1−4 (η)

2q 12 (q 2 − 1)

x14 = x2−4 (1)x3+4 (1)x1−2 (−1)x1−3 (−1)

24q 12

x15 = x2−4 (1)x3+4 (1)x1−2 (−η)x1−3 (−1)

8q 12

x16 = x2−4 (1)x2+4 (−η)x1−2+3+4 (1)x1−3 (−1)

4q 12

x17 = x2−4 (1)x3+4 (1)x1−2−3+4 (1)x1−2 (−η)x1−3 (ξ)

4q 12

x18 = x2 (1)x3+4 (1)x1−2+3−4 (1)x1−2 (−1)x1−3 (ζ)

3q 12

x19 x2−3 (1)x3−4 (1)x4 (1)

q 8 (q 2 − 1)

x20 = x2 (1)x3+4 (1)x1−2−3−4 (1)

q 8 (q 2 − 1)

x21 = x2−4 (1)x3 (1)x2+4 (1)x1−2−3+4 (1)

2q 8

x22 = x2−4 (1)x3 (1)x2+4 (η)x1−2−3+4 (1)

2q 8

x23 = x2−3 (1)x3−4 (1)x4 (1)x1−2 (1)

2q 6

x24 = x2−3 (1)x3−4 (1)x4 (1)x1−2 (η)

2q 6

x25 = x2−3 (1)x3−4 (1)x4 (1)x1−2−3−4 (1)

3q 4

x26 = x2−3 (1)x3−4 (1)x4 (1)x1−2−3−4 (1)x1−2+3+4 (ζ)

3q 4

x27 = x2−3 (1)x3−4 (1)x4 (1)x1−2−3−4 (1)x1−2+3+4 (−ζ) 3q 4

Lemma 3.2.8. Let u ∈ G be an element of order 3, where G = E6 (3t ) or G = 2 E6 (3t ) is a canonical finite group of Lie type. Then u is conjugate to u−1 in G. ′

Proof. Let G and σ be such that G = Op (G). Since the characteristic equals 3, we have that Z(Gsc ) = 1. So one can assume G = Gsc to be universal. Thus, G is simply connected and G = Gσ . We assembled in Table 6 the information from [32, Lemmas 4.2, 4.3, 4.4, and Theorem 4.13] on conjugacy classes of unipotent elements of G. In Table 6, we substitute the root α1 r1 + α2 r2 + α3 r3 + α4 r4 + SIBERIAN ADVANCES IN MATHEMATICS

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α5 r5 + α6 r6 , where r1 , r2 , r3 , r4 , r5 , r6 form a fundamental system of E6 , by the 6-tuple α1 α2 α3 α4 α5 α6 of its coefficients. Note that if n > 3 and r1 , r2 , . . . , rn are fundamental roots of a root system of type An then |xr1 (1)xr2 (1) . . . xrn (1)| > 3. Using this fact, we obtain that |x4 | > 3, |x7 | > 3, |x8 | > 3, and |xi | > 3, hr (λ)

where i > 10, i 6= 12, 16. Thus we need to consider the remaining cases only. We have that x1 1 = x−1 1 , where λ is a square root of −1 in F3 . For each x ∈ G, denote by Ccl(x) its conjugacy class in G. Since CG (x1 ) = CG (x1 )0 , from [26, Theorem 8.5] we have that, for every Frobenius map σ and for every x ∈ Ccl(x1 ) ∩ Gσ , the elements x and x−1 are conjugate under Gσ . So, if x ∈ Ccl(x1 ) ∩ G then x is conjugate to its inverse. Table 6. Unipotent classes in E6 (F3 ) representative x

C = CG (x) |C : C 0 |

x1 = x10000 (1)

1

x2 = x100000 (1)x001000 (1)

2

x3 = x100000 (1)x000100 (1)

1

x4 = x100000 (1)x001000 (1)x000100 (1)

1

x5 = x100000 (1)x001000 (1)x000010 (1)

1

x6 = x100000 (1)x000100 (1)x000001 (1)

1

x7 = x100000 (1)x001000 (1)x000100 (1)x000010 (1)

1

x8 = x100000 (1)x001000 (1)x000100 (1)x000001 (1)

1

x9 = x100000 (1)x001000 (1)x000010 (1)x000001 (1)

1

x10 = x100000 (1)x001000 (1)x010000 (1)x000010 (1)

1

x11 = x100000 (1)x001000 (1)x000100 (1)x010000 (1)x000001 (1)

1

x12 = x100000 (1)x001000 (1)x000010 (1)x000001 (1)x010000 (1)

1

x13 = x100000 (1)x001000 (1)x000100 (1)x000010 (1)x000001 (1)

1

x14 = x010000 (1)x001000 (1)x000100 (1)x000010 (1)

1

x15 = x010000 (1)x001000 (1)x000100 (1)x010110 (1)

6

x16 = x000001 (1)x000010 (1)x001000 (1)x010000 (1)

1

x17 = x010000 (1)x001000 (1)x000010 (1)x101100 (1)

1

x18 = x000010 (1)x000100 (1)x001000 (1)x100000 (1)x000001 (1)x111111 (1)

2

x19 = x010000 (1)x000100 (1)x000010 (1)x000001 (1)x101000 (1)x001110 (1)

1

x20 = x100000 (1)x010000 (1)x001000 (1)x000100 (1)x000010 (1)x000001 (1)

3

For all the other xi such that |xi | = 3 and i 6= 2, we act in the same way. It remains to consider x2 . By [26, Theorem 8.5], we have that, for every Frobenius map σ, Ccl(x2 ) ∩ Gσ consists of two conjugacy classes of G = Gσ . First, assume that G = E6 (3t ). Then, by [32, Lemmas 4.2 and 4.4], we have that if x ∈ Ccl(x2 ) ∩ G then x is conjugate in G either to y1 = x100000 (1)x001000 (1) or to y2 = x100000 (1)x001000 (1)x000001 (1)x122321 (η), where η is a nonsquare in F3t . By [32, Lemma 4.2], we have |CG (y1 )| = 2q 26 (q 2 − 1)2 (q 3 − 1)2 , and by [32, Lemma 4.4], |CG (y2 )| = 2y 26 (q 4 − 1)(q 6 − 1). For i = 1, 2, let CclG (yi ) be the conjugacy class of yi in G. Since |CG (y1 )| = 6 |CG (y2 )|, we have that yi SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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is conjugate to its inverse in G for i = 1, 2. So, if x ∈ CclG (y1 ) or x ∈ CclG (y2 ) then x is conjugate to its inverse in G. Now, assume that G = 2 E6 (3t ) and denote E6 (32t ) by G1 . Then G = (G1 )τ for some graph-field automorphism τ of G1 . There exists a Frobenius map σ such that G1 = Gσ , G = Gστ (see [22, (7-2)]). Let Ccl1 and Ccl2 be two conjugacy classes of G1 contained in Ccl(x2 ) ∩ G1 . We prove that every x ∈ Ccli , i = 1, 2, is conjugate to x−1 in G1 . Since Ccl(x2 ) ∩ G consists of two conjugacy classes of G, we have that Ccl1 ∩ G consists of one conjugacy class and Ccl2 ∩ G consists of one conjugacy class as well. So, every x ∈ Ccli ∩ G, i = 1, 2 is conjugate to its inverse under G. ′

Lemma 3.2.9. Let Op (Gσ ) ≤ G ≤ Gσ be a finite adjoint group of Lie type over a field of odd characteristic p, and the root system Φ of G is one of the following: An (n > 2), Dn (n > 4), Bn (n > 3), G2 , F4 , E6 , E7 or E8 ; and G 6≃ 2 G2 (32n+1 ). Let U be a maximal unipotent subgroup of G, let H be a Cartan subgroup of G normalizing U , and let Q be a Sylow 2-subgroup of H. Then CU (Q) = {e}. ′

′

Proof. Clearly, it suffices to prove the lemma for the case G = Op (Gσ ) = Op (G), i. e., we can assume that G is a canonical adjoint group of Lie type. First, assume that GQis split. Assume that CU (Q) 6= {e} and u ∈ CU (Q) \ {e}. Consider decomposition (1) of u = r∈Φ+ xr (tr ), where tr are from the definition field Fq of G. In view of [5, Theorem 5.3.3, (ii)], this decomposition is unique. Since, for every h(χ) ∈ H, r ∈ Φ, t ∈ Fq , the formula h(χ)xr (t)h(χ)−1 = xr (χ(r)t) holds (see [5, p. 100]) then we obtain that each multiplier xr (tr ) in decomposition (1) of u is in CU (Q). So we can assume that u = xr (t) for some r ∈ Φ+ and t ∈ F∗q . Under our restriction on Φ, by the Hartley–Shute lemma 1.5.5, there exists h(χ) ∈ H such that χ(r) = −1. Since h(χ)2 = h(χ2 ) (see [5, p. 98]), we have that χ2 (r) = 1, i. e., |h(χ)2 | < |h(χ)|. Hence, |h(χ)| is even and we may write h(χ) = h2 · h2′ = h(χ1 ) · h(χ2 ), a decomposition of h(χ) as a product of its 2- and 2′ parts. Further, χ(r) = χ1 (r) · χ2 (r), therefore, χ1 (r) = −1 and χ2 (r) = 1. Thus, h(χ1 )xr (t)h(χ1 )−1 = xr (−t) 6= xr (t). Since h(χ1 ) ∈ Q, the obtained enequality contradicts the choice of xr (t) ∈ CU (Q). Assume that G ≃ 2 An (q), G ≃ 2 Dn (q), or G ≃ 2 E6 (q). Then Φ(G) equals An , Dn and E6 , respectively. Denote by r¯ the image of r of Φ under the corresponding symmetry. In terms of [5], the root system Φ(G) is expressible as a union of the equivalency classes Ψi , where each Ψi Q has one of the following types: A1 , A1 × A1 , or A2 . In view of [5, Proposition 13.6.1], the equality U = i XΨi holds, where XΨi = {xr (t) | t ∈ Fq }

if Ψi = {r} has the type A1 (here r = r¯); XΨi = {xr (t)xr¯(tq ) | t ∈ Fq2 } if Ψi = {r, r¯} has the type A1 × A1 (here r 6= r¯, and r + r¯ 6∈ Φ(G)); XΨi = {xr (t)xr¯(tq )xr+¯r (u) | t ∈ Fq2 , u + uq = −Nr,¯r ttq } if Ψi = {r, r¯, r + r¯} has the type A2 (here r 6= r¯ and r + r¯ ∈ Φ(G)). Further, if h(χ) is an element of H then the following equalities hold (see [5, p. 263]): h(χ)xr (t)h(χ)−1 = xr (χ(r)t) if r = r¯ and Ψi = {r} has the type A1 ; h(χ)xr (t)xr¯(tq )h(χ)−1 = xr (χ(r)t)xr¯(χ(¯ r )tq ) if r 6= r¯, r + r¯ 6∈ Φ(G) and Ψi = {r, r¯} has the type A1 × A1 ; h(χ)xr (t)xr¯(tq )xr+¯r (u)h(χ)−1 = xr (χ(r)t)xr¯(χ(¯ r )tq )xr+¯r (χ(r + r¯)u) if r = 6 r¯, r + r¯ ∈ Φ(G) and Ψi = {r, r¯, r + r¯} has the type A2 . Let u be a nontrivial element from CU (Q). Then u contains aQnontrivial multiplier from XΨi for some i. In view of the uniqueness of decomposition into the product i XΨi (see [5, Proposition 13.6.1]), we can assume that u ∈ XΨ . SIBERIAN ADVANCES IN MATHEMATICS

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Assume that Ψ has the type A1 , i. e., u = xr (t), t ∈ Fq , r = r¯. In view of the Hartley–Shute lemma 1.5.5, for each s ∈ Fq , there exists h(χ) ∈ H such that χ(r) = s. Take s = −1. Then there exists h(χ) ∈ H such that χ(r) = −1. Since h(χ)2 = h(χ2 ) (see the formula on p. 98 in [5]), we have that χ2 (r) = 1, i. e., |h(χ)2 | < |h(χ)|. Hence, the order |h(χ)| is even and we may write h(χ) = h2 · h2′ = h(χ1 ) · h(χ2 ), i. e., as a decomposition of h(χ) into the product of its 2- and 2′ -parts. Further, χ(r) = χ1 (r) · χ2 (r), therefore, χ1 (r) = −1 and χ2 (r) = 1. Thus, h(χ1 )xr (t)h(χ1 )−1 = xr (−t) 6= xr (t). So the case when u = xr (t) and Ψ = {r} has the type A1 , is impossible. Assume that Ψ = {r, r¯} has the type A1 × A1 . By the Hartley–Shute lemma 1.5.5, for every s ∈ Fq2 , there exists h(χ) ∈ H such that χ(r) = s2 . Since there exists s ∈ Fq2 such that s2 = −1, there exists h(χ) ∈ H such that χ(r) = −1. As above, h(χ) can be written as h(χ1 ) · h(χ2 ), i. e., as a product of its 2- and 2′ - parts. Then χ1 (r) = −1, and so, h(χ1 )xr (t)xr¯(tq )h(χ1 )−1 = xr (−t)xr¯(−tq ) 6= xr (t)xr¯(tq ). Thus the case when u = xr (t)xr¯(tq ) and Ψ = {r, r¯} has the type A1 × A1 , is impossible. Assume that Ψ = {r, r¯, r + r¯} has the type A2 . By Hurtley-Shute lemma 1.5.5, for each s ∈ Fq2 there exists h(χ) ∈ H such that χ(r) = s3 . Choose s = −1 then there exists h(χ) ∈ H such that χ(r) = −1. Again h(χ) = h(χ1 ) · h(χ2 ) is expressible as a product of its 2- and 2′ - parts and χ1 (r) 6= 1. Then h(χ1 )xr (t)xr¯(tq )xr+¯r (u)h(χ1 )−1 = xr (−t)xr¯(χ1 (−tq )xr+¯r (χ1 (r + r¯)u) 6= xr (t)xr¯(tq )xr+¯r (u) for t 6= 0. If t = 0 then choose s so that s2 = −1. Then χ1 (r + r¯) = −1 and, as above, we obtain the inequality. Hence this case is impossible. Finally, assume that G ≃ 3 D4 (q). In terms from [5], a root system Φ(G) is expressible as a union of the equivalency classes Ψi , when each Q Ψi has the type either A1 , or A1 × A1 × A1 . In view of [5, Proposition 13.6.1], the equality U = i XΨi holds, where XΨi = {xr (t) | t ∈ Fq }

if Ψi = {r} has the type A1 (here r = r¯); 2

XΨi = {xr (t)xr¯(tq )xr¯(tq ) | t ∈ Fq3 } if Ψi = {r, r¯, r¯} has the type A1 × A1 × A1 (here r 6= r¯ and r + r¯ 6∈ Φ(G)). In both the cases, by the Hartley–Shute lemma 1.5.5, there exists h(χ) ∈ H such that χ(r) = −1. As above, we can assume that h(χ) is a 2-element, i. e., h(χ) ∈ Q and h(χ) does not centralizes nonidentical elements from XΨi , and, in the last case, from here it follows the statement. Lemma 3.2.10. In the notation of Lemma 3.2.9, with p odd, let K be a Carter subgroup of G such that |K| = 2a pb . Then a > 0. More precisely, up to conjugation, Op (K) ≤ CU (Q). In particular, under the assumptions of Lemma 3.2.9, K is a 2-group. Proof. The condition a = 0 implies K = U . But U is normalized by H which is non-trivial as p is odd and G is simple. Thus a > 0. Now, assume b > 0. By the Borel-Tits theorem (Lemma 1.5.4), K is contained in a proper parabolic subgroup P of G and Op (K) ≤ Op (P ). Since P = LOp (P ), where L is a Levi factor of P , from Lemma 2.4.1 it follows that KOp (P )/Op (P ) ∼ = O2 (K) is a Carter subgroup of P/Op (P ) ∼ = L. Thus, O2 (K) is a Sylow 2-subgroup of L. But L contains H, therefore we can assume that Q ≤ K. From here, it follows that Op (K) ≤ CU (Q). Lemma 3.2.11. Let G be a non-Abelian simple group not of Lie type. Then every element z of odd order is conjugate to some z k 6= z. Proof. By the classification of finite simple groups, G is either alternating or sporadic. Our claim can be verified directly in the first case, and in the second one, by the description of the conjugacy classes given in [10]. SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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3.3. Almost simple groups which are not minimal counter examples. In this Subsection, A denotes a minimal almost simple group that is a minimal counter example (see definition 2.4.7). If G is a group of Lie type then we denote by Field(G) the subgroup of Aut(G) generated by the inner, diagonal, and field b = G to unify the notation. automorphisms. If G is a simple group which is not of Lie type then we set G Moreover, for each x ∈ G, we assume that the composition factors of the centralizer CG (s) are known simple groups, and so, CG (s) satisfies (C). As we noted in Subsection 2.4, this assumption is always valid. We say it here in order to emphasize that all the results below do not depend on the classification of finite simple groups. Lemma 3.3.1. Let A be a minimal counter example and G = F ∗ (A). Assume that, for every b of odd prime order, z is conjugate to some z k 6= z in G. Then A is not a minimal element z ∈ G counter example if one of the following holds: b ∩ A| is a 2-power; (a) |A : G

b ∩ A)|2′ > 1; b : (G b ∩ A)| is a 2-power and if Φ(G) has the type D4 then |(Field(G)∩) : (G (b) |G (c) for every odd prime r and every Sylow r-subgroup R of A, either R ∩ G has no complement in R, or all such complements are conjugate in A. Proof. Let K and H be nonconjugate Carter subgroups of A. Note that by Lemma 2.4.2 (b), it b and H ∩ G b are 2-groups. We first prove (c) and next we show that (a) and (b) follows follows that K ∩ G from (c). (c) By Theorem 2.1.4 and Lemma 2.4.1, we obtain that KG/G = HG/G = A/G. In particular, if r is a prime divisor of |A/G| then r divides both |K| and |H|. By Lemma 2.4.2 and by the conditions b and H ∩ G b do not contain elements of odd prime order, i. e., they of this lemma, it follows that K ∩ G are 2-groups. If R ∩ G has no complement in R then we get a contradiction immediately. If all such complements are conjugate in A then we obtain a contradiction with Lemma 2.4.2 (c). Thus we obtain that |A/G| is a 2-power, and hence, K and H are 2-groups that is impossible. Further, (a) evidently follows from (c). As to (b), then it also follows from (c) by the conjugacy of complements that follows by Lemma 4.2.6. Note that all non-Abelian composition factors of the centralizer of every element of the alternating group Altn are alternating groups of lower degree. So, Lemmas 3.2.11 and 3.3.1 and induction on n imply immediately that the Carter subgroups of Aut(Altn ), with n > 5, either are Sylow 2-subgroups or do not exist. The same statement holds for the sporadic groups. Thus, the following assertion is valid. Lemma 3.3.2. Let S be a finite non-Abelian simple group which is either sporadic or alternating. Then, in every subgroup A of Aut(S), a Carter subgroup either does not exist or is a Sylow 2-subgroup. (pt ), with Theorem 3.3.3. Let G be a finite adjoint group of Lie type such that G = PΩ± 2(2ℓ+1) ℓ > 2. Then G is not a minimal counter example. Proof. Assume that our statement is not valid. Then G contains a Carter subgroup K which is not a 2-group. Let s ∈ Z(K) be an element of odd prime order r. Then we can assume that s is semisimple except probably the case when p 6= 2 and |K| = 2a pb . But this is impossible in view of Lemmas 3.2.9 and 3.2.10. Hence, s is semisimple, and from the relation K ≤ CG (s) it follows that CG (s) is self-normalizing t in G (see Lemma 2.4.2 (a)). Now, let G = Ω2(2ℓ+1) (Fp ) and let σ be such that Gσ = Ω± 2(2ℓ+1) (p ).

Moreover, set K0 to be equal to the preimage of K in Gσ . Clearly, K0 is a Carter subgroup of Gσ and we may identify s with its preimage in Gσ since the center of Gσ has the order 2 or 4. Since |s| is odd, Lemma 3.2.1 implies that C = CG (s) is a connected reductive subgroup of maximal rank of G (see Lemmas 1.5.1 and 1.5.2). Moreover, C is a proper subgroup of G since s ∈ / Z(G). By Lemma 3.2.4 the group NW (WC )/WC is isomorphic to NG (C)/C. Using Lemma 3.2.4 and the description of NW (WC )/WC given in [7, Proposition 10], we conclude that NG (CG (s))/CG (s) is trivial only if WC⊥ and AutW (∆C ) are both trivial. From assumption ℓ ≥ 2 it follows that this occurs precisely when m1 = 0 and m2ℓ+1 = 1 (in the notation of [7]). In this case, C = A2ℓ (Fp ) ◦ S, where S is a 1-dimensional torus. Using the fact that G contains exactly one class of connected reductive subgroups isomorphic to C, and assuming that SIBERIAN ADVANCES IN MATHEMATICS

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 0 I , we may identify C with the image of GL2ℓ+1 (Fp ) G preserves the bilinear form induced by J =  I 0 under a monomorphism ϕ such that   A 0  A 7→  . 0 (A−1 )t

By the Lang-Steinberg theorem (Lemma 1.5.3), we can assume that either Cσ = ϕ(GL2ℓ+1 (pt )) or Cσ = ϕ(GU2ℓ+1 (p2t )). Since K0 is a Carter subgroup of Cσ and ℓ ≥ 2, by [16], [17], and Theorem 1.5.6, it follows that K0 is the normalizer of a Sylow 2-subgroup P of Cσ , and either pt = 2 (and Cσ = ϕ(GL2ℓ+1 (pt ))) or p is odd. From s ∈ Z(Cσ ) it follows that r = |s| divides pt − 1 if Cσ ≃ GL2ℓ+1 (pt ), and that r divides pt + 1 if Cσ ≃ GU2ℓ+1 (p2t ). In particular, p is odd. Using the known structure of normalizers of Sylow 2-subgroups in classical groups (see [28] and [9]), we can assume that K0 is a subgroup of the following group   ϕ   B 0  | B ∈ GL2ℓ (pt ) , β ∈ F∗q if Cσ ≃ GL2ℓ+1 (pt ) L=    0 β   ϕ  B 0   | B ∈ GU2ℓ (p2t ) , β pt +1 = 1 if Cσ ≃ GU2ℓ+1 (p2t ). L=   0 β  ϕ  I2ℓ 0  , where γ has order r. Since y is in the As we noted above, there exists y ∈ L such that y =  0 γ center of L, it is also in the center of K0 . Thus, K0 ≤ CCGσ (s) (y) = CC (y) σ . From the isomorphism rank C ≃ GL2ℓ+1 (Fp ) it follows that CC (y) is a connected reductive σ-invariant subgroup of maximal of G. Thus, in view of the above mentioned result by R. Carter [7, Proposition 10], CC (y) σ is selfnormalizing in Gσ only if CC (y) is conjugate to C. But dim(CC (y)) < dim(C) since y is not in the center of C. Thus, (CC (y))σ is not self-normalizing in Gσ . Since Z(G) ≤ CC (y), it follows that the factor group (CC (y))σ /(Z(G))σ is not self-normalizing in Gσ /(Z(G))σ = G. Thus we obtained a contradiction with Lemma 2.4.2 (a) since K is contained in (CC (y))σ /(Z(G))σ and (CC (y))σ /(Z(G))σ satisfies (C). ε t Theorem 3.3.4. Let E6ε (pt ) ≤ G ≤ E\ 6 (p ). Then G is not a minimal counter example.

Proof. Assume that our claim is not valid. Then, by Lemma 2.4.2 (c), G admits a Carter subgroup K which does not contain any Sylow 2-subgroup of G. In particular, K is not a 2-group. Let s ∈ Z(K) have an odd prime order r. By Lemmas 3.2.5, 3.2.8, and 2.4.2, p does not divide |K|. Hence, s is semisimple and K is contained in CG (s) which, in virtue of Lemma 2.4.2 (a), is self-normalizing. If |s| = 6 3 then, by Lemma 3.2.2, it follows that CG (s) is connected. If |s| = 3 then, by Lemma 1.5.2, it follows that |C : C 0 | divides ∆ = 3. Direct calculations by using [13] and [32] show that CG (s) is not self-normalizing if |s| = 3. Therefore we can assume that |s| = 6 3 and CG (s) is connected. Since CG (s) is self-normalizing, Lemma 3.2.4 shows that C = CG (s) is self-normalizing as well. By [32], we obtain that C is self-normalizing if and only if C = A4 (Fp ) ◦ A1 (Fp ) ◦ S or C = D5 (Fp ) ◦ S, where S is a 1dimensional torus of G. If C = A4 (Fp ) ◦ A1 (Fp ) ◦ S then, as in the proof of Theorem 3.3.3, we can find an element y ∈ Z(K) such that |y| = r and CG (hsi × hyi) is not self-normalizing; a contradiction with Lemma 2.4.2. So, assume that C = D5 (Fp ) ◦ S. Then CG (s) = C ∩ G = HL, where H is a Cartan subgroup of G ′ ˆ : L| divides 4 then and L = Op (CG (s)) is either D5 (pt ) or 2 D5 (pt ). Since |L O2′ (H) = (O2′ (H) ∩ Z(CG (s))) × (O2′ (H) ∩ L). SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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Denoting by Q a Sylow 2-subgroup of CG (s), we claim that NCG (s) (Q) = QZ(CG (s)). Indeed, let x be an element of NCG (s) (Q). From H = O2 (H) × O2′ (H) and CG (s) = HL it follows that we can write x = h1 zl, where h1 ∈ O2 (H), z ∈ O2′ (H) ∩ Z(CG (s)), l ∈ L. Clearly, we can assume that O2 (H) ≤ P , thus, l ∈ NCG (s) (P ). Since L is normal in CG (s), it implies that l ∈ NL (P ∩ L). By [28], NL (P ∩ L) = P ∩ L, and hence, l ∈ P . We conclude that NCG (s) (P ) = P Z(CG (s)) is nilpotent. Hence it is a Carter subgroup of CG (s). Since CG (s) < G, the Carter subgroups of CG (s) are conjugate. Therefore, up to conjugation, M = NCG (s) (P ). In virtue of the formula |(C)σ | = |Mσ | · |(Z(C)0 )σ |, where Mσ = L in our notation (cf. [7]), we have that |G : CG (s)| is odd. So, P is a Sylow 2-subgroup of G, a contradiction. Our results are summarized in the following Theorem 3.3.5. An almost simple group A with socle G is not a minimal counterexample in the following cases: (a) G is alternating, sporadic or one of the following groups: A1 (pt ), Bℓ (pt ), and Cℓ (pt ), where t is even if p = 3; 2 B2 (22n+1 ), G2 (pt ), F4 (pt ), 2 F4 (22n+1 ), and 3 D4 (q 3 ); E7 (pt ), where p 6= 3; E8 (pt ), where p 6= 3, 5; b ∩ A| is a 2-power or G is one of the following groups: D2ℓ (pt ), 3 D4 (p3t ), and 2 D2ℓ (p2t ), (b) |A : G b ∩ A)|2′ > 1; where t is even if p = 3, and moreover, if G = D4 (pt ) then |(Field(G) ∩ A) : (G c)

A = G is one of the following groups: Bℓ (3t ), Cℓ (3t ), D2ℓ (3t ), 2 D2ℓ (32t ), D2ℓ+1 (pt ), E6 (r t ), 2 E6 (r 2t ), E7 (3t ), E8 (3t ), and E8 (5t ).

2D 2t 3 3t 2 2n+1 ), 2ℓ+1 (r ), D4 (3 ), G2 (3

In particular, none simple group except possibly groups isomorphic to Aεn (q), cannot be a minimal counterexample. Moreover, if every almost simple group with a known simple normal subgroup satisfies (C) then, in the above mentioned groups, a Carter subgroup (if it exists) contains a Sylow 2-subgroup. b of odd prime order is conjugate in G with some Proof. (a) and (b). We claim that every element z ∈ G k z 6= z. When G is alternating or sporadic, this is valid by Lemma 3.2.11, and when G is of Lie type and z semisimple, this is valid by Lemma 3.2.3. On the other hand, when z is unipotent (so p is odd), our claim follows from Lemmas 3.2.6, 3.2.7 if K = G2 (3t ), F4 (3t ) and from Lemma 3.2.5 in the remaining cases. Finally, if G ≃ 3 D4 (q 3 ) then, by [40, Theorem 1.2 (vi)], every element of G is conjugate to its inverse. Since we do not consider unipotent elements in bad characteristic, these items follow from Lemma 3.3.1 b : G| is a 2-power or |A : A ∩ G| b is a since, for all groups under consideration, we have that either |G 2-power (for example, cf. [10]). (c) Our claim follows from [17] (see Theorem 1.5.6) when G = B2 (3t ) ≃ C2 (3t ) or G = Cℓ (3t ), and from Theorems 3.3.3 and 3.3.4 when G is one of the groups D2ℓ+1 (pt ), 2 D2ℓ+1 (p2t ), E6 (pt ), or 2 E6 (p2t ). b of odd order is So, assume that one of the remaining cases is valid. Every semisimple element z ∈ G −1 conjugate with z by Lemma 3.2.3. Thus, in characteristic 2, a Carter subgroup M of G can only be a Sylow 2-subgroup, and in odd characteristic, M can only have order 2a pb . If G 6= 2 G2 (32n+1 ) then the assumptions of Lemma 3.2.9 are satisfied and, by Lemma 3.2.10, we conclude that M is again a 2-group. Now, assume G = 2 G2 (32n+1 ) (n > 0) and let z be unipotent element of order 3 in the center of M . By [47, Chapter III], there is an element y 6= z in G such that y and z are conjugate in G and CM (y) = CM (z). From y = z x we have (CG (z))x = CG (y) = CG (z), hence, x ∈ CG (z) by Lemma 2.4.2(b) (in this case, even though 3 is a bad prime, the structure of centralizers of all elements is known and their composition factors are known simple groups; so we do not use CFSG here). Thus, y = z, a contradiction. We conclude that M is a 2-group in this case as well. Note that after proving that, for any known finite simple group S and a nilpotent subgroup N ≤ Aut(S), the Carter subgroups of h N, S i are conjugate, Theorem 3.3.5 implies that Carter subgroups of the groups mentioned in the theorem, must contain a Sylow 2-subgroup. By Lemma 2.4.3, this is possible if and only if the normalizer of a Sylow 2-subgroup P of A satisfies NA (P ) = P CA (P ), i. e., if and only if A satisfies (ESyl2). In [28] and the results below, all the almost simple groups satisfying (ESyl2) are completely classified. Thus the classification of Carter subgroups in groups from Theorem 3.3.5 is known. SIBERIAN ADVANCES IN MATHEMATICS

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4. SEMILINEAR GROUPS OF LIE TYPE In this Section, we define semilinear groups of Lie type and transfer structure results for groups of Lie type to semilinear groups of Lie type. This tool is very important in the classification of Carter subgroups in groups of Lie type extended by field, graph, or graph-field automorphisms given in Section 5. 4.1. Basic definitions. We define some overgroups of finite groups of Lie type. First, we give a precise description of a Frobenius map σ. Note that all the maps in this Section are automorphisms if G is considered as an abstract group, and they are endomorphisms if G is considered as an algebraic group. Since we use the maps to construct connected automorphisms in finite groups and groups over an algebraically closed field, in this Section, we find it convenient to call all the maps automorphisms. Let G be a connected simple linear algebraic group over an algebraically closed field Fp of positive characteristic p. In the sequel, unless otherwise stated, we consider groups of adjoint type. Choose a − Borel subgroup B of G, let U = Ru (B) be the unipotent radical of B. There exists a Borel subgroup B , − with B ∩ B = T , where T is a maximal torus of B (so of G). Here we partially repeat some notation and definitions from Subsection 1.3. Let Φ be the root system of G and let {X r | r ∈ Φ+ } be the set of T invariant 1-dimensional root subgroups of U . Each X r is isomorphic to the additive group of Fp . Hence every element of Xr can be written as xr (t), where t is the image of xr (t) under the above mentioned − − − isomorphism. Denote by U = Ru (B ) the unipotent radical of B . As above, define T -invariant 1− − dimensional subgroups {X r | r ∈ Φ− } of U . Then G = hU , U i. Let ϕ¯ be a field automorphism of G (as of an abstract group) and γ¯ be a graph automorphism of G. It is known that the automorphism ϕ¯ may be chosen so that it acts by the rule xr (t)ϕ¯ = xr (tp ) (for example, see [5, 12.2] and [8, 1.7]). In view of [5, Propositions 12.2.3 and 12.3.3], we can choose γ¯ so that it acts by the rule xr (t)γ¯ = xr¯(t) if Φ has no roots of distinct length, and by the rule xr (t)γ¯ = xr¯(tλr ) for suitable λr ∈ {1, 2, 3} if Φ has roots of distinct length. Recall that r¯ is the image of r under the symmetry ρ (corresponding to γ¯ ) of a root system Φ. In both the cases, we can write xr (t)γ¯ = xr¯(tλr ), where λr ∈ {1, 2, 3}. From this ∗ formulas it follows that ϕ¯ · γ¯ = γ¯ · ϕ. ¯ Let nr (t) = xr (t)x−r (−t−1 )xr (t) and N = hnr (t) | r ∈ Φ, t ∈ Fp i. ∗ Let hr (t) = nr (t)nr (−1) and H = hhr (t) | r ∈ Φ, t ∈ Fp i. By [5, Chapters 6 and 7], H is a maximal torus of G, N = NG (H), and the subgroups X r are root subgroups with respect to H. So we can ¯ and γ¯ -invariant under our choice. Moreover, ϕ¯ induces a substitute T by H and assume that T is ϕtrivial automorphism of N /H. The automorphism ϕ¯k , with k ∈ N, is called a classical Frobenius automorphism. We will call an automorphism σ a Frobenius automorphism if σ is conjugate under G, with γ¯ǫ ϕ¯k , ǫ ∈ {0, 1}, k ∈ N. By the Lang-Steinberg theorem (Lemma 1.5.3), it follows that, for every g¯ ∈ G, the elements σ and σ¯ g are conjugate under G. Thus, by [38, 11.6], we have that a Frobenius map defined in Subsection 1.4 coinsides with a Frobenius automorphism defined here. ¯ γ¯ , and σ = γ¯ ǫ ϕ¯k . Assume that |¯ γ | 6 2, i. e., we do not consider a triality automorphism Now, fix G, ϕ, of a group G with the root system Φ(G) = D4 . Set B = B σ , H = H σ , and U + = U σ . Since B, H, and U are ϕ¯ and γ¯ -invariant, they give us respectively a Borel subgroup, a Cartan subgroup, and a maximal unipotent subgroup (a Sylow p-subgroup) of Gσ (for more detail, see [8, 1.7–1.9] or [23, Chapter 2]). ′

Assume that ǫ = 0, i. e., Op (Gσ ) is not twisted (is split). Then U + = hXr | r ∈ Φ+ i, where Xr is isomorphic to the additive group of Fpk = Fq , and each element of Xr can be written as xr (t), t ∈ Fq . −

Set also U − = U σ . As for U + , we may write U − = hXr | r ∈ Φ− i and each element of Xr can be written as xr (t), t ∈ Fq . Now we can define the automorphism ϕ as the restriction of ϕ¯ on Gσ and the automorphism γ as the restriction of γ¯ on Gσ . By definition, the equalities xr (t)ϕ = xr (tp ) and xr (t)γ = xr¯(tλr ) hold for all r ∈ Φ (see the definition of γ¯ above). Define the automorphism ζ of Gσ by ζ = γ ε ϕℓ , ϕℓ 6= e, ε ∈ {0, 1}, and the automorphism ζ¯ of G by ζ¯ = γ¯ ε · ϕ¯ℓ . Choose a ζ-invariant ′ subgroup G, with Op (Gσ ) ≤ G ≤ Gσ . Note that if the root system Φ of G is not equal to D2n then ′ Gσ /(Op (Gσ )) is cyclic. Thus for most groups and automorphisms except groups of type D2n over a SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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′

field of odd characteristic, every subgroup G of Gσ , with Op (Gσ ) ≤ G ≤ Gσ , is γ- and ϕ-invariant. Define ΓG as the set of subgroups of type hG, ζgi ≤ Gσ ⋋ hζi, where g ∈ Gσ , hζgi ∩ Gσ ≤ G, and ΓG ¯ Following [23, Definition 2.5.13], the automorphism ζ is called as the set of subgroups of type G ⋋ hζi. a field automorphism if ε = 0, i. e., ζ = ϕℓ , and is called a graph-field automorphism in the remaining cases (recall that we assume that ϕℓ 6= e). −

′

Now, assume that ǫ = 1, i. e., Op (Gσ ) is twisted. Then U + = U σ and U − = U σ . Define ϕ on U ± ′ ′ as the restriction of ϕ¯ on U ± . Since Op (Gσ ) = hU + , U − i, the map ϕ is an automorphism of Op (Gσ ). ′ Consider ζ = ϕℓ 6= e, and let G be a ζ-invariant group with the property Op (Gσ ) ≤ G ≤ Gσ . Then ζ¯ = ϕ¯ℓ is an automorphism of G. Define ΓG as the set of subgroups of the type hG, ζgi ≤ Gσ ⋋ hζi, ¯ Following [23, where g ∈ Gσ , hζgi ∩ Gσ ≤ G; and ΓG as the set of subgroups of the type G ⋋ hζi. Definition 2.5.13], we say that ζ is a field automorphism if |ζ| is not divisible by |γ| (this definition is used also in the case when |γ| = 3 and Gσ ≃ 3 D4 (q)), and that ζ is a graph automorphism in the remaining cases. The groups from the above-defined set ΓG are called semilinear finite groups of Lie type (they ′ are also called semilinear canonical finite groups of Lie type if G = Op (Gσ )) while the groups from the set ΓG are called semilinear algebraic groups. Note that ΓG cannot be defined without ΓG since ′ we need to know that ϕℓ 6= e. If G is written in the notation from [5], i. e., Op (G) = G = An (q) or ′ Op (G) = G = 2 An (q), etc., then we will write ΓG as ΓAn (q), Γ2 An (q), etc. Consider A ∈ ΓG and x ∈ A \ G. Then x = ζ k y for some k ∈ N and y ∈ Gσ . Define x ¯ to be equal to k ζ¯ y. Conversely, if x ¯ = ζ¯k y for some y ∈ Gσ , ζ k 6= e, and hζ k yi ∩ Gσ ≤ G, define x to be equal to ζ k y. ¯ = ∞. If x ∈ G, set x ¯ = x. Note that we need not to assume that x ¯∈ / G since |ζ| Lemma 4.1.1. In the above notation, consider a subgroup X of G. An element x normalizes X if and only if x ¯ normalizes X as a subgroup of G. Proof. Since ζ is a restriction of ζ¯ on G, our statement is evident. Let X1 be a subgroup of A ∈ ΓG. Then X1 is generated by a normal subgroup X = X1 ∩ G and by an ¯ element of the form x = ζ k y. By Lemma 4.1.1, we may consider the subgroup X 1 = h¯ x, Xi of G ⋋ hζi. Now we find it reasonable to explain why we use so complicated notations and definitions. We have that the order of ζ is always finite but the order of ζ¯ is always infinite. Thus, even if Z(G) is trivial, we cannot consider G ⋋ hζi as a subgroup of Aut(G). Therefore we need to define in a some way (one possible way is just given) a connection between the elements from Aut(G) and the elements from Aut(G) in order to use the technique of linear algebraic groups. Let R be a σ-stable maximal torus (a reductive subgroup of maximal rank or a parabolic subgroup, respectively) of G, and let an element y ∈ NG⋋hζi ¯ (R) be chosen so that there exists x ∈ hG, ζgi such that y = x ¯. Then R1 = hx, R ∩ Gi is called a maximal torus (a reductive subgroup of maximal rank or a parabolic subgroup, respectively) of hG, ζgi. 4.2. Translation of basic results. Lemma 4.2.1. Let M = hx, Xi, where X = M ∩ G E M is a subgroup of hG, ζgi such that Op (X) is nontrivial. Then there exists a proper σ- and x ¯-invariant parabolic subgroup P of G such that X ≤ P and Op (X) ≤ Ru (P ). Proof. Define U0 = Op (X), N0 = NG (U0 ), and by induction, Ui = U0 Ru (Ni−1 ) and Ni = NG (Ui ). Clearly, Ui and Ni are x ¯- and σ-invariant for all i. By [25, Proposition 30.3], the chain of subgroups N0 ≤ N1 ≤ . . . ≤ Nk ≤ . . . is finite and P = ∪i Ni is a proper parabolic subgroup of G. Clearly, P is σand x ¯-invariant, X ≤ P , and Op (X) ≤ Ru (P ). ′

Lemma 4.2.2. Let Op (Gσ ) ≤ G ≤ Gσ be a finite adjoint group of Lie type with a base field ′ of characteristic p and order q. Assume also that Op (G) is not isomorphic to 2 D2n (q), 3 D4 (q), 2 B (22n+1 ), 2 G (32n+1 ), and 2 F (22n+1 ). Then there exists a maximal σ-stable torus T of G such 2 2 4 that SIBERIAN ADVANCES IN MATHEMATICS

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(a) (NG (T )/T )σ ≃ (NG (T ))σ /(T σ ) = N (Gσ , T σ )/T σ ≃ W , where W is the Weyl group of G; (b) if r is an odd prime divisor of q − (ε1), where ε = + if G is split, and ε = − if G is twisted, then N (Gσ , T σ ) contains a Sylow r-subgroup Gσ ; (c) if r is a prime divisor of q − (ε1), and s is an element of order r of G such that CG (s) is connected then, up to conjugation by an element of G, an element s is contained in T = T σ ∩ G; ′

If a prime r in item (b) exists then a torus T satisfying (a) − (c) is unique in Op (Gσ ) up to conjugation, and |T σ | = (q − ε1)n , where n is a rank of G. ′

Proof. Since, for every maximal torus T of Gσ , the equality Gσ = T Op (Gσ ) holds, we can assume that G = Gσ without lost of generality. If G is split then the lemma can be easily proven. In this case, T is a maximal torus such that T σ is a Cartan subgroup of Gσ (i. e., T is a maximal split torus) and (a) is evident. Item (b) follows from [22, (10.1)]. Moreover, from [22, (10.2)] it follows that the order of T σ is uniquely defined and is equal to (q − 1)n , where n is a rank of G. By [1, F, §6], we have that each element of order r of T is contained in Gσ . Now there exists g ∈ G such that sg ∈ T , hence, sg ∈ G. In view of connectedness of the centralizer of s, the elements s and sg are conjugate in G if and only if they are conjugate in G. So, s and sg are conjugate in G, whence it follows (c). The information about classes of maximal tori in [1, G] and [4] implies that, up to conjugation by an element from G, there exists a unique torus T such that |T σ | = (q − 1)n . ′

Assume that Op (G) ≃ 2 An (q). Then T is a maximal torus such that |T σ | = (q + 1)n . Note that T σ can be obtained from a maximal split torus by twisting by the element w0 σ. Direct calculations by using [8, Proposition 3.3.6] show that N (Gσ , T σ )/T σ is isomorphic to W (G) which in turn is isomorphic to Symn+1 . The uniqueness follows from [7, Proposition 8]. Item (b) follows from [22, (10.1)]. As to item (c), we first show that each element of order r from T is in G. Assume that t is an element of order r in T (recall that, in this case, r divides q + 1). Let H be a σ-stable maximal split torus of G. The torus T σ is obtained from H by twisting by w0 σ, where w0 ∈ W (G) is a unique element mapping all the positive roots into the negatives, and T σ ≃ H σw0 . Let r1 , . . . , rn be the set of fundamental roots of An . Then t as an element of H, can be written as hr1 (λ1 ) · . . . · hrn (λn ). Now, for each i, we have σw0 : hri (λ) 7→ h−ri (λq ) = hri (λ−q ), i. e., tσw0 = t−q . Since r divides q + 1, we obtain that tq+1 = e, i. e., t = t−q . Hence, tσw0 = t and t ∈ T σ . Now, as in the nontwisted case, there exists g ∈ G such that sg ∈ T , therefore, sg ∈ T σ . In view of connectedness of CG (s), the elements s and sg are conjugate in G. ′

For Op (G) = 2 D2n+1 (q), we take T to be equal to a unique (up to conjugation in G) maximal torus which has the order |T σ | = (q + 1)2n+1 (the uniqueness follows from [7, Proposition 10]), and ′ for Op (G) = 2 E6 (q), we take T to be equal to a unique (again up to conjugation in G) maximal torus which has the order |T σ | = (q + 1)6 (the uniqueness follows from [13, Table 1, p. 128]). As in the case of G = 2 An (q), it is easy to show that T satisfies items (a), (b), and (c) of the lemma. ′

Lemma 4.2.3. Let G be a finite group of Lie type and G and σ are chosen so that Op (Gσ ) ≤ G ≤ Gσ . Let s be a regular semisimple element of odd prime order r of G. Then NG (CG (s)) 6= CG (s). Proof. In view of [26, Proposition 2.10], we have that CG (s)/CG (s)0 is isomorphic to a subgroup of ∆(G). Now, if the root system Φ of G is not equal to either An or E6 then |∆(Φ)| is a power of 2. Since ∆(G) is a quotient of ∆(Φ(G)), Lemma 3.2.1 implies that CG (s) = CG (s)0 = T is a maximal torus and CG (s) = CG (s) ∩ G = T . Since NG (T ) ≥ N (G, T ) 6= T , we obtain the statement of the lemma in this case. Thus we can assume that either Φ = An or Φ = E6 . ′ Assume first that Φ = An , i. e., Op (G) = Aεn (q), where ε ∈ {+, −}. Clearly, T = CG (s)0 ∩ G is a normal subgroup of CG (s), hence, CG (s) ≤ N (G, T ). Assume that NG (CG (s)) = CG (s). Then CG (s) = NN (G,T ) (CG (s)) and CG (s)/T is a self-normalizing subgroup of N (G, T )/T . As noted above, CG (s)/T is isomorphic to a subgroup of ∆(An ), i. e., it is cyclic. By Lemma 3.2.1, we also have that CG (s)/T is an r-group. Therefore, CG (s)/T = hxi for some r-element x ∈ N (G, T )/T . Thus, hxi is SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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a Carter subgroup of N (G, T )/T . Now, in view of [8, Proposition 3.3.6], we have that N (G, T )/T ≃ CW (G) (y) for some y ∈ W (G) ≃ Symn+1 . Clearly, CCW (G) (y) (x) contains y. Thus, y must be an relement, otherwise NCW (G) (y) (hxi) contains an element of order coprime to r, i. e., NCW (G) (y) (hxi) 6= hxi. We obtained a contradiction with the fact that hxi is a Carter subgroup of CW (G) (y). Let y = τ1 · . . . be the decomposition of y into the product of independent cycles, and l1 , . . . be the lengths of τ1 , . . ., respectively. Assume first that m1 cycles has the same length l1 , m2 cycles has the length l2 , etc. Let m0 = n + 1 − (l1 m1 + . . . + lk mk ). Then CSymn+1 (y) ≃ Zl1 ≀ Symm1 × . . . × Zlk ≀ Symmk × Symm0 ,

where Zli is a cyclic group of order li . If mj > 1 for some j > 0 then there exists a normal subgroup N of CSymn+1 (y) such that CSymn+1 (y)/N ≃ Symmj 6= {e}. By Lemma 3.3.2, the Carter subgroups in every group S, with Altℓ ≤ S ≤ Symℓ , are conjugate for all ℓ > 1. Thus CW (G) (y) and N satisfy (C), and hxi is a unique Carter subgroup of CW (G) (y) up to conjugation. By Lemma 2.4.1, we obtain that hxi maps onto a Carter subgroup of CW (G) (y)/N ≃ Symmj . By Lemma 3.3.2, only a Sylow 2-subgroup of Symmj can be a Carter subgroup of Symmj . We obtained a contradiction with the fact that x is an r-element and r is odd. Thus we can assume that CW (G) (y) ≃ (Zl1 × . . . × Zlk ) and li 6= lj if i 6= j. From the known structure of maximal tori and their normalizers in Aεn (q) (for example, see [7, Propositions 7,8]) we obtain the structure of T and N (G, T ) which we explain below using a matrix technique. In the sequel, the group GLεn (q) is isomorphic to GLn (q) if ε = + and is isomorphic to GUn (q) if ε = −. For the decomposition l1 + l2 + . . . + lk = n + 1, consider a subgroup L of GLεn+1 (q) consisting of blockdiagonal matrices of the type   A 0 . . . 0  1     0 A2 . . . 0   ,    ... ... ... ...    0 0 . . . Ak

where Ai ∈ GLεli (q). Then L ≃ GLεl1 (q) × . . . × GLεlk (q). Denote, for brevity, GLεli (q) by Gi . In every group Gi , consider a Singer cycle Ti . It is well known that NGi (Ti )/Ti is a cyclic group of order li ′ and N (Gi , Ti ) = NGi (Ti ). There exists a subgroup Z of Z(SLεn+1 (q)) such that Op (G) ≃ SLεn+1 (q)/Z. Then T ≃ (T1 × . . . × Tk ) ∩ SLεn+1 (q) /Z, N (G, T ) ≃ (N (G1 , T1 ) × . . . × N (Gk , Tk )) ∩ SLεn+1 (q) /Z.

Since, for every Singer cycle Ti , the group N (Gi , Ti )/Ti is cyclic, we can assume that N (G, T ) = CG (s) n+1 −(ε1)n+1 and n + 1 = r k for some k > 1 (the and T is a Singer cycle, i. e., is a cyclic group of order q q−(ε1) k

last equality holds since N (G, T )/T is an r-group). But q r ≡ q (mod r), hence, r divides q − (ε1). By Lemma 4.2.2, we obtain that s is in N (G, H), where H is a maximal torus such that the factor group N (G, H)/H is isomorphic to Symn+1 and |H| = (q − ε1)n . In particular, H is not a Singer cycle. If s ∈ H, this immediately implies a contradiction with the choice of s. If s 6∈ H then since the order of s is prime, the intersection hsi ∩ H is trivial. Hence, under the natural homomorphism N (G, H) → N (G, H)/H ≃ Symn+1 , the element s is mapped on an element of order r. But in Symn+1 , every element of odd order is conjugate to its inverse. Thus, there exists a 2-element z of G which normalizes but not centralizes hsi. Therefore, z ≤ NG (CG (s)) ≤ NG (CG (s)0 ) and |N (G, T )/T | is divisible by 2 that contradicts the above proven statement that N (G, T )/T is an r-group. This contradiction completes the proof in the case Φ(G) = An . SIBERIAN ADVANCES IN MATHEMATICS

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In the remaining case Φ = E6 , it is easy to see that, for every y ∈ W (E6 ), the group CW (E6 ) (y) does not contain a Carter subgroup of order 3. Indeed, if CW (E6 ) (y) has a Carter subgroup of order 3 then it is generated by y. But it is known (and can be easily verified by [4, Table 9]) that, in W (E6 ), there is no elements of order 3 whose centralizer has order 3. Since |CG (s)/T | divides 3 and the group CG (s)/T is a Carter subgroup of CW (E6 ) (y) for some y, a contradiction. By Lemma 4.2.3, we obtain the similar result for semilinear groups: Lemma 4.2.4. Let hG, ζgi be a finite semilinear group of Lie type, and let G and σ be chosen ′ so that Op (Gσ ) ≤ G ≤ Gσ . Let s be a regular semisimple element of odd prime order of G. Then NhG,ζgi (ChG,ζgi (s)) 6= ChG,ζgi (s). Proof. Since s is semisimple, there exists a σ-stable maximal torus S of G containing s. Since ′ Gσ = Op (Gσ )S σ , we can assume that g ∈ S σ , i. e., the elements g and s commute. If ChG,ζgi (s)G 6= hG, ζgi then we can substitute hG, ζgi by ChG,ζgi (s)G and prove the lemma for this group. Moreover, if ChG,ζgi (s) = CG (s) then the assertion follows from Lemma 4.2.3, so we can assume that ζ centralizes s. If either G is not twisted or |ζ| is odd then, by [23, Proposition 2.5.17], it follows that we can assume σ = ζ¯k for some k > 0. By Lemma 4.2.3, there exists an element of NGζg (CG (s)) not contained in CGζg (s) and the assertion follows. Now, assume that G is twisted and |ζ| is even. Then σ = γ¯ ϕ¯k , ζ¯ = ϕ¯ℓ , where k divides ℓ. Therefore s is in Gγ¯ . In dependence of the root system Φ(G), we obtain that Gγ¯ is isomorphic to a simple algebraic group with the root system equal to Bm (for some m > 1), Cm (for some m > 2), or F4 . By Lemma ′ 3.2.3, the element s is conjugate to its inverse under Op ((Gγ¯ )σζg ¯ ) ≤ Gζg , so, NhG,ζgi (ChG,ζgi (s)) 6= ChG,ζgi (s). Lemma 4.2.5. Let G be a finite group of Lie type over a field of odd characteristic p. Assume ′ that G and σ are chosen so that Op (Gσ ) ≤ G ≤ Gσ . Let ψ be a field automorphism of odd order ′ of Op (Gσ ). Then ψ centralizes a Sylow 2-subgroup of G, and there exists a ψ-stable Cartan subgroup H such that ψ centralizes a Sylow 2-subgroup of H. Moreover, if G 6≃ 2 G2 (32n+1 ), 3 D (q 3 ), 2 D (q 2 ) then there exists a ψ-stable torus T of G such that ψ centralizes a Sylow 24 2n subgroup of T and the factor group N (G, T )/T is isomorphic to NG (T )/T . Proof. Clearly, we need to prove the lemma only for the case G = Gσ . Assume that |ψ| = k. Let Fq be the base field of G. Then q = pα and α = k · m. Now |G| can be written as |G| = q N (q m1 + ε1 1) · . . . · (q mn + εn 1) for some N , where n is the rank of G, εi = ± (see [5, Theorems 9.4.10 and 14.3.1]). Similarly, we have that |Gψ | = (pm )N ((pm )m1 + ε1 1) · . . . · ((pm )mn + εn 1), i. e., |G|2 = |Gψ |2 and a Sylow 2-subgroup of Gψ is a Sylow 2-subgroup of G. By [23, Proposition 2.5.17], there exists an automorphism ψ1 of G such that σ = ψ1k and ψ coincides with the restriction of ψ1 on Gσ . Note that, in general, ψ1 is not equal to ψ¯ defined above. Consider a maximal split torus H ψ1 of Gψ1 . Then H = H σ is a ψ-stable Cartan subgroup of G. Since |H| = (q k1 + ε1) · . . . · (q kl + εl 1), where εi = ±, the equality |H|2 = |Hψ |2 can be proven in the same way. Now, assume that G 6≃ 2 G2 (32n+1 ), 3 D4 (q 3 ), 2 D2n (q 2 ). By Lemma 4.2.2, there exists a maximal torus T of Gψ such that N (Gψ , T )/T ≃ NG (T )/T and |Tψ | = (pm − ε1)n . Since |ψ| is odd and T ψ1 is obtained from a maximal split torus H by twisting by an element w0 then T σ is also obtained from a maximal split torus H by twisting by element w0 (see the proof of Lemma 4.3.1). Therefore |T σ | = (q − ε1)n , |T ψ1 | = (pm − ε1)n , hence, |T σ |2 = |T |2 = |Tψ |2 . Lemma 4.2.6. [22, (7-2)] Let G be a connected simple linear algebraic group over a field of characteristic p, and let σ be a Frobenius map of G and G = Gσ be a finite group of Lie type. Let ϕ be a field or a graph-field automorphism of G (if ϕ is graph-field then corresponding graph automorphism has order 2) and let ϕ′ be an element of (G ⋋ hϕi) \ G such that |ϕ′ | = |ϕ| and ′ G ⋋ hϕi = G ⋋ hϕ′ i. Then there exists g ∈ G such that hϕig = hϕ′ i. In particular, if G/Op (G) is a ′ 2-group and ϕ is of odd order then g can be chosen in Op (G). SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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A particular case of the following lemma is proven in [21, Theorem A]). Lemma 4.2.7. Let G be a finite adjoint split group of Lie type, and let G and σ be chosen so ′ ′ that Op (Gσ ) ≤ G ≤ Gσ . Assume that τ is a graph automorphism of order 2 of Op (G). Then every ′ semisimple element s ∈ G is conjugate to its inverse under hOp (Gσ ), τ ai, where a is an element of Gσ . Proof. If Φ(G) is not of type An , D2n+1 , E6 then the assertion follows from Lemma 3.2.3. Thus we need to consider the groups of type An , D2n+1 , E6 . Denote by τ¯ a graph automorphism of G such that τ¯|G = τ . Let T be a maximal σ-stable torus of G such that T σ ∩ G is a Cartan subgroup of G. Let r1 , . . . , rn be fundamental roots of Φ(G) and ρ be the symmetry corresponding to τ¯. Denote riρ by r¯i . Then T = hhri (ti ) | 1 6 i 6 n, ti 6= 0i and hri (ti )τ¯ = hr¯i (ti ). Denote by W the Weyl group of G. Let w0 be a unique element of W mapping all positive roots onto negative roots and let n0 be its preimage in NG (T ) under the natural homomorphism NG (T ) → NG (T )/T ≃ W . Since σ acts trivially on W = N (G, T )/T (see Lemma 4.2.2), we can take n0 ∈ G, i. e., nσ0 = n0 . Then, for all ri and t, we have that hri (t)n0 τ¯ = hrw0 ρ (t) = h−ri (t) = hri (t−1 ). i

xn0 τ¯

x−1

Thus, = for all x ∈ T . Now, let s be a semisimple element of G. Then there exists a maximal σ-stable torus S of G containing g s. Since all the maximal tori of G are conjugate, we have that there exists g ∈ G such that S = T . ′ −1 Since Gσ = Op (Gσ )T σ , we can assume that a ∈ T σ . Therefore sgn0 τ¯ag = s−1 . Since nσ0 = n0 and τ¯σ = τ¯, we have that (gn0 τ¯ag−1 )σ = gσ n0 τ¯a(g−1 )σ . Moreover, since S is σ-stable then, for every x ∈ σ −1 σ −1 S, we have that xgn0 τ¯ag = xg n0 τ¯a(g ) = x−1 , i. e., gn0 τ¯ag−1 S = gσ n0 τ¯a(g−1 )σ S. In particular, there exists t ∈ S such that gn0 τ¯ag−1 t = gσ n0 τ¯a(g−1 )σ . In view of the Lang-Steinberg Theorem (Lemma 1.5.3), there exists y ∈ S such that t = y · (y −1 )σ . Therefore, gn0 τ¯ag−1 y = (gn0 τ¯ag−1 y)σ , i. e., −1 ′ gn0 τ ag−1 y ∈ Gσ ⋋ hτ i, and sgn0 τ ag y = s−1 . Since Op (Gσ )S σ = Gσ and S σ is Abelian, we may find ′ z ∈ S σ such that gn0 τ ag−1 yz ∈ hOp (Gσ ), τ ai. 4.3. Carter subgroups of special type. In this Subsection, we consider some problems on the existence and structure of Carter subgroups in semilinear groups containing a Sylow 2-subgroup, or from the normalizer of a Borel subgroup. Lemma 4.3.1. Let G be a finite group of Lie type over a field of odd characteristic, and let G ′ and σ be chosen so that Op (Gσ ) ≤ G ≤ Gσ . If G satisfies (ESyl2) then every subgroup L, with G ≤ L ≤ Gσ , satisfies (ESyl2). ′

Proof. Let Q be a Sylow 2-subgroup of Gσ and Q0 = Op (Gσ ) ∩ Q be a Sylow 2-subgroup of ′ Op (Gσ ). Clearly, if NGσ (Q0 ) = QCGσ (Q) then the statement of the lemma is valid. In view of [28, Theorem 1], for a classical group Gσ , the equality NGσ (Q0 ) = QCGσ (Q) may be not valid if the root system of G has the type A1 or Cn only. If the root system of G has the type A1 or Cn then ′ |Gσ : Op (Gσ )| = 2 and the statement of the lemma follows from Lemma 2.4.6. ′

Assume now that G is a group of exceptional type. If Gσ = Op (Gσ ) then the statement of the lemma is clearly true. The equality NGσ (Q0 ) = QCGσ (Q) may be not valid if the root system of G has the type ′ E6 or E7 only. If the root system of G has the type E7 then |Gσ : Op (Gσ )| = 2 and the statement of the lemma follows from Lemma 2.4.6. ′ ′ Assume that the root system of G has the type E6 . Then either Gσ = Op (Gσ ) or |Gσ : Op (Gσ )| = 3. ′ In the first case, we have nothing to prove, so assume that |Gσ : Op (Gσ )| = 3. Since the group G ′ coincides either with Gσ or with Op (Gσ ), and since, in the case G = Gσ , there is nothing to prove, we ′ can assume that G = Op (Gσ ). By [23, Theorem 4.10.2], there exists a maximal torus T of Gσ such that Q is contained in N (Gσ , T ). Since |Gσ : G| = 3 then Q = Q0 ≤ N (G, T ∩ G). By [29, Theorem 6], the SIBERIAN ADVANCES IN MATHEMATICS

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equality NG (Q) = Q × R0 holds, where R0 ≤ T is a cyclic group of odd order. Now since Gσ = T G then NGσ (Q) = hNT (Q), NG (Q)i. Indeed, N (G, T ∩ G)/(T ∩ G) ≃ N (G, T )/T . Hence, a Sylow 2subgroup QT /T of N (G, T )/T coincides with its normalizer. Since the factor group Gσ /G is cyclic of order 3 then NGσ (Q) = htg, NG (Q)i, where t ∈ T and g ∈ G. Moreover, since |Gσ : G| = 3, we can assume that tg is an element of order 3k for some k > 0. Since t ∈ T ≤ N (Gσ , T ) then Qt ≤ −1 N (G, T ∩ G). So there exists an element g1 ∈ N (G, T ∩ G) such that Qt = Qg1 . Therefore we can assume that tg = tg1 ∈ N (Gσ , T ). Under the natural epimorphism π : N (Gσ , T ) → N (Gσ , T )/T , the image of NN (Gσ ,T ) (Q) coincides with Q. Hence, (tg)π = e, so tg ∈ T . Thus each element of odd order of Gσ normalizing Q is in T . Since T is a torus then T is Abelian. Hence the set of elements of odd order of NGσ (Q) forms a normal subgroup R ≤ T . Therefore, NGσ (Q) = Q × R, i. e., Gσ satisfies (ESyl2). The following lemma is immediate consequence of [28, Theorem 1]. ′

Lemma 4.3.2. Let Op (Gσ ) = G be a canonical finite group of Lie type and let G be either of type A1 or of type Cn , and let q = pα be the order of the base field of G, where p is odd. Then G satisfies (ESyl2) if and only if q ≡ ±1 (mod 8). Note that Lemma 4.3.1 together with [28, Theorem 1] and [29, Theorem 6] implies that every group of Lie type over a field of odd characteristic distinct from a Ree group and groups from Lemma 4.3.2, satisfies (ESyl2). Lemma 4.3.3. Let G be a finite adjoint group of Lie type over a field of odd characteristic, ′ G 6≃ 3 D4 (q 3 ), and let G and σ be chosen so that Op (Gσ ) ≤ G ≤ Gσ . Let A be a subgroup of ′ ′ Aut(Op (Gσ )) such that A ∩ Gσ = G. If Op (G) ≃ D4 (q), assume also that A is contained in the group generated by the inner-diagonal and field automorphisms, and a graph automorphism of order 2 as well. Then A satisfies (ESyl2) if and only if G satisfies (ESyl2). Proof. Assume that G satisfies (ESyl2). Under the conditions of the lemma, we have that the factor group A/G is Abelian, so, A/G = A1 × A2 , where A1 is a Hall 2′ -subgroup of A/G and A2 is a Sylow 2-subgroup of A/G. Denote by A1 the complete preimage of A1 in A. If A1 satisfies (ESyl2) then, by Lemma 2.4.6, A satisfies (ESyl2) as well. Thus we can assume that the order |A/G| is odd. Since we are assuming that a graph automorphism of order 3 is not contained in A then A/G is cyclic, hence, A = hG, ψgi, where ψ is a field automorphism of odd order and g ∈ Gσ . Since |A : G| = |ψ| is odd, we can assume that |ψg| is also odd. By Lemma 4.2.5, ψ centralizes a Sylow 2-subgroup of Gσ , therefore g is of odd order. Further, the quotient Gσ /G is Abelian and can be written as L × Q, where L is a Hall 2′ -subgroup of Gσ /G and Q is a Sylow 2-subgroup of Gσ /G. Let L be the complete preimage of L in Gσ under the natural homomorphism. Then g ∈ L. Consider L ⋋ hψi ≥ A. By construction, |L ⋋ hψi : A| = |L : G| is odd. By Lemma 4.3.1, the group L satisfies (ESyl2). By Lemma 4.2.5, the field automorphism ψ centralizes a Sylow 2-subgroup Q of L. Thus, NL⋋hψi (Q) = NL (Q) × hψi = QCL (Q) × hψi = QCL⋋hψi (Q), i. e., the group L ⋋ hψi satisfies (ESyl2). Since |L ⋋ hψi : A| is odd then A satisfies (ESyl2). Now, assume that A satisfies (ESyl2). If G does not satisfies (ESyl2) then the corresponding results from [28, Theorem 1] and [29, Theorem 6] imply that the root system of G either has the type A1 or ′ has the type Cn . In both the cases, the factor group Aut(Op (Gσ )/Gσ ) is cyclic and is generated by a field automorphism ϕ. Further, from [28, Theorem 1], it follows that the order of the base field (which coincides with the field of definition in this case since G is not twisted) is equal to q = pt and q ≡ ±3 ′ (mod 8). Therefore, t is odd, and so, |Aut(Op (Gσ ))/Gσ | is odd as well. Thus, |A : G| is odd. Hence, G satisfies (ESyl2). Lemma 4.3.4. Let hG, ζgi be a finite semilinear group of Lie type over a field of characteristic p (we do not exclude the case hG, ζgi = G) and G is of adjoint type (recall that g ∈ Gσ but not necessarily g ∈ G). Assume that B = U ⋋ H, where H is a Cartan subgroup of G, is a ζginvariant Borel subgroup of G, and hB, ζgi contains a Carter subgroup K of hG, ζgi. Assume that K ∩ U 6= {e}. Then one of the following statements holds: SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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\ 2t (a) either hG, ζgi = h2 A2 (22t ), ζgi or hG, ζgi = 2 A 2 (2 ) ⋋ hζi; the order |ζ| = t is odd and is not \ 2 divisible by 3, CG (ζ) ≃ 2 A 2 (2 ), and K ∩ G is Abelian and has order 2 · 3; (b) hG, ζgi = h2 A2 (22t ), ζgi, the order |ζ| = t is odd, CG (ζ) ≃ 2 A2 (22 ), and the subgroup K ∩ G is a Sylow 2-subgroup of Gζ ; 2t (c) either hG, ζgi = hA2 (22t ), ζgi or hG, ζgi = A\ 2 (2 ) ⋋ hζi, where ζ is a graph-field automor\ 2 phism of order 2t, with t not divisible by 3, CG (ζ) ≃ 2 A 2 (2 ), and the subgroup K ∩ G is

Abelian and has the order 2|ζ2′ | · 3; (d) hG, ζgi = hA2 (22t ), ζgi, where ζ is a graph-field automorphism and CG (ζ) ≃ 2 A2 (22 ), and K ∩ G is a Sylow 2-subgroup of Gζ2′ ; (e) G is defined over F2t , hG, ζgi = G ⋋ hζgi, where ζ is either a field automorphism of order t ′ ′ ′ of O2 (G) if O2 (G) is split, or a graph automorphism of order t if O2 (G) is twisted, and K = Q ⋋ hζgi up to conjugation in G, where Q is a Sylow 2-subgroup of G(ζg)2′ ; (f) G is split and defined over F2t , hG, ζgi = G ⋋ hζgi, where ζ is a product of a field auto′ morphism of odd order t of O2 (G), and a graph automorphism of order 2, ζ and ζg are conjugate under Gσ , and K = Q ⋋ hζgi up to a conjugation in G, where Q is a Sylow 2subgroup of G(ζg)2′ ; (g) G/Z(G) ≃ PSL2 (3t ), the order |ζ| = t is odd (hence, ζ ∈ hG, ζgi), and K contains a Sylow 3-subgroup of Gζ3′ ; (h) hG, ζgi = 2 G2 (32n+1 ) ⋋ hζi, |ζ| = 2n + 1, and K ∩ 2 G2 (32n+1 ) = Q × P , where Q is of order 2 and |P | = 3|ζ|3 . Note that, in all items (a)–(h) of the lemma, the Carter subgroups having the indicated structure, exist. The existence of Carter subgroups in items (a) and (c) follows from the existence of a Carter subgroup of order 6 in PGU3 (2) (see [17]). The existence of Carter subgroups in items (b) and (d)(f) follows from the fact that a Sylow 2-subgroup in a group of Lie type defined over a field of order 2, coincides with its normalizer. The existence of Carter subgroups in item (g) follows from the fact that a Sylow 3-subgroup of PSL2 (3) coincides with its normalizer. The existence of a Carter subgroup satisfying item (h) of the lemma, follows from the existence of a Carter subgroup K of order 6 in a (non simple) group 2 G2 (3). The existence of a Carter subgroup K of order 6 in 2 G2 (3) follows from the results in [30] and [47]. Proof. If G is one of the groups A1 (q), G2 (q), F4 (q), 2 B2 (22n+1 ), or 2 F4 (22n+1 ) then the assertion follows from Table 3. If hG, ζgi = G then the assertion follows from the results of Section 3 and Theorem 1.5.6. So we can assume that hG, ζgi = 6 G, i. e., that ζ is a nontrivial field, graph-field, or graph automorphism. If Φ(G) = Cn , the lemma follows from Theorem 5.2.3 below, that does not use Lemma 4.3.4, so we assume that Φ(G) 6= Cn . If Φ(G) = D4 and either a graph-field automorphism ζ is a product of a field automorphism and a graph automorphism of order 3, or G ≃ 3 D4 (q 3 ) then the lemma follows from Theorem 5.3.1 below which does not use Lemma 4.3.4, so we assume that hG, ζgi is contained in the group A1 defined in Theorem 5.3.1, and G 6≃ 3 D4 (q 3 ). Since we will use Lemma 4.3.4 in the proof of Theorem 5.4.1, after Theorems 5.2.3 and 5.3.1, it is possible to make such additional assumptions. Assume that q is odd and Φ(G) is one of the following types: An (n > 2), Dn (n > 4), Bn (n > 3), E6 , E7 , or E8 . By Lemma 2.4.1, we have that KU/U is a Carter subgroup of hB, ζgi/U ≃ hH, ζgi. Since Gσ = GH σ , where H is a maximal split torus of G and H σ ∩ G = H, then we can assume that g ∈ H σ , In particular, g centralizes H. So, Hζ ≤ Z(hH, ζgi) and we obtain that Hζ ≤ K up to conjugation in B. By Lemma 4.2.5, the automorphism ζ2′ centralizes a Sylow 2-subgroup Q of H. Thus, each element of odd order of hH, ζgi centralizes Q and Lemma 2.4.3 implies that the inclusion Q ≤ K holds up to conjugation in B. By Lemma 3.2.9, it follows that CU (Q) = {e}, a contradiction with the fact that K ∩ U is nontrivial. ′ Assume that G ≃ 2 G2 (32n+1 ) and hG, ζgi = G ⋋ hζi (in this case Op (Gσ ) = Gσ ). Using Lemma 2.4.1 once more, we have that KU/U is a Carter subgroup of (B ⋋ hζi)/U ≃ H ⋋ hζi. By Lemma SIBERIAN ADVANCES IN MATHEMATICS

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3.2.3, every semisimple element of G is conjugate to its inverse. Since non-Abelian composition factors of every semisimple element of G can be isomorphic only to groups A1 (q), by Table 3, it follows that the centralizer of every semisimple element of G satisfies condition (C). So Lemma 2.4.2 implies that KU/U ∩ B/U is a 2-group. On the other hand, |H|2 = 2 and KU/U ≥ Z(B/U ) ≥ Hζ , hence, |Hζ | = 2 and |ζ| = 2n + 1. Thus, K ∩ G = (K ∩ U ) × hti, where t is an involution. From here it follows that K ∩ U = CG (t) ∩ Gζ3′ . Now the structure results from [30, Theorem 1] and [47] imply item (h) of the lemma. Now, let q = 2t . Assume first that Φ(G) has one of the types An (n > 2), Dn (n > 4), Bn (n > 3), E6 , E7 , or E8 , and G is split and ζ is a field automorphism. As above, we obtain that Hζ ≤ K, ′ and O2 (Gζ ) is a split group of Lie type with the definition field of order q = 2t/|ζ| . By the Hartley– ′ Shute Lemma 1.5.5, for every r ∈ Φ(G) and every s ∈ GF (2t/|ζ| )∗ , there exists h(χ) ∈ Hζ ∩ O2 (Gζ ) t such that χ(r) = s. The same arguments as in Lemma 3.2.9 imply that if |ζ| 6= 1 then the inequality K ∩ U ≤ CU (Hζ ) = {e} holds, a contradiction. So, |ζ| = t and Hζ = {e}. Since g can be chosen in H σ and hζgi ∩ Gσ ≤ hζgi ∩ H σ ≤ Hζ = {e} then hζgi ∩ Gσ = {e}. By Lemma 4.2.6, the elements ζg and ζ are conjugate under Gσ , and item (e) of the lemma follows. Now, assume that Φ(G) is of type An (n > 3), Dn (n > 4), or E6 , and either ζ is a graph-field automorphism and G is split or G is twisted. Let ρ be the symmetry of the Dynkin diagram of Φ(G) corresponding to γ (recall that ζ = γ ε ϕℓ by definition), and r¯ denotes r ρ for r ∈ Φ(G). As above, it is possible to prove that Hζ ≤ K up to conjugation. If Hζ 6= {e} then, by the Hartley–Shute Lemma 1.5.5, we obtain that CU (Hζ ) = {e} which contradicts the condition K ∩ U 6= {e}. If Hζ = {e} then either G is twisted and |ζ| = t that implies statement (e) of the lemma, or G is twisted, |ζ| = 2t, and in particular, t is odd that implies item (f) of the lemma. ′

6 2t then Assume that O2 (G) ≃ A2 (2t ), ζ is a graph-field automorphism, and t is odd. If |ζ| = arguments similar to the proof of Lemma 2.4.2 and the Hartley–Shute Lemma 1.5.5, prove the relation CU (Hζ ) = {e} which contradicts the condition K ∩ U 6= {e}. If |ζ| = 2t then we obtain item (f) of the lemma. ′ Assume now that O2 (G) ≃ A2 (22t ) and ζ is a graph-field automorphism. Using once more the Hartley–Shute Lemma 1.5.5 for |ζ| = 6 2t, we again obtain the relation CU (Hζ ) = {e} which contradicts 2 2 \ 2 the condition K ∩ U 6= {e}. If |ζ| = 2t then either Gζ ≃ 2 A2 (22 ) or Gζ ≃ 2 A 2 (2 ). If Gζ ≃ A2 (2 ) \ 2 then Hζ = {e} and we obtain the statement (d) of the lemma. If Gζ ≃ 2 A 2 (2 ) then |Hζ | = 3 and so, KU/U ∩ HU/U is a cyclic group hyi of order (2t3 + 1)3 = 3k , where 3k−1 = t3 . If k > 1 then the Hartley–Shute Lemma 1.5.5 implies that CU (y) = {e} that is impossible. Thus, t is not divisible by 3 and K ∩ U is contained in the centralizer of an element x generating Hζ . Consider the homomorphism GL3 (22t ) → PGL3 (22t ). Then some preimage of x is similar to the matrix   λ 0 0    0 λ2 0  ,   0 0 λ where λ is the generating element of the multiplicative group of GF (22 ). The preimage of U is similar to the set of upper triangular matrices with the same elements on the diagonal. Direct calculations show that CU (x) is isomorphic to the additive group of GF (22t ). The nilpotency of K implies that K ∩ U = (CU (x))ζ2′ , and item (c) of the lemma follows. ′

Assume now that O2 (G) ≃ 2 A2 (22t ). By Lemma 2.4.1, KU/U is a Carter subgroup of hB, ζgi/U ≃ hH, ζgi, and as above, we can assume that Hζ ≤ K. If |ζ| = 2t then Gζ ≃ SL3 (2) and Hζ = {e}, and ′ item (e) of the lemma follows. Assume that t is even and |ζ| 6 t. Then either O2 (Gζ ) ≃ SL3 (22t/|ζ| ) ′ (if the order |ζ| is even) or O2 (Gζ ) ≃ 2 A2 (22t/|ζ| ) (if the order |ζ| is odd, hence, |ζ| < t). Clearly, Hζ contains an element x such that K ∩ U ≤ CU (Hζ ) = {e} which is a contradiction with the condition SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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K ∩ U 6= {e}. If t is odd and t 6= |ζ| then O2 (Gζ ) ≃ 2 A2 (22t/|ζ| ), and from here it follows that Hζ contains an element x such that CU (x) = {e}. If |ζ| = t and t is odd then the order |KU/U ∩ B/U | can be divisible only by 3 (otherwise by the Hartley–Shute Lemma 1.5.5 it again follows that CU (Hζ ) = (22t/|ζ| ) then {e}). If G ≃ 2 A (22t/|ζ| ) then H = {e} and we obtain item (b) of the lemma. If G ≃ 2 A \ ζ

2

ζ

ζ

2

KU/U ∩ HU/U is a cyclic group hyi of order (2t3 + 1)3 = 3k , where 3k−1 = t3 . If k > 1 then Hartley– Shute Lemma 1.5.5 implies the relation CU (y) = {e} that is impossible. Thus, t is not divisible by 3 and K ∩ U is contained in the centralizer of an element x generating Hζ . As in the non-twisted case above, we obtain that CU (x) is isomorphic to the additive group of GF (2t ). The nilpotency of K implies that K ∩ U = (CU (x))ζ2′ , and item (a) of the lemma follows. 5. CARTER SUBGROUPS OF SEMILINEAR GROUPS 5.1. Brief review of the results . In this Section, using notation and results obtained in Section 4, we classify the Carter subgroups in groups of automorphisms of finite groups of Lie type. First we give such a classification in the case when a group of Lie type has the type Cn or when a group of its automorphisms contains a triality automorphism since the arguments in these two cases differ from those in the other ones. Then we formulate the final theorem and we prove it in the two subsequent Subsections. In the last Subsection, we prove that, in every finite group with known composition factors, the Carter subgroups are conjugate. 5.2. Carter subgroups of symplectic groups . Consider a set A of almost simple groups A such that a unique non-Abelian composition factor S = F ∗ (A) is a canonical simple group of Lie type and A contains nonconjugate Carter subgroups. If the set A is not empty, denote by Cmin the minimal possible order of F ∗ (A), with A ∈ A. If the set A is empty then let Cmin = ∞. We will prove that Cmin = ∞, i. e., that A = ∅. Note that if A ∈ A and G = F ∗ (A) then there exists a subgroup A1 of A such that A1 ∈ A and A1 = KG for a Carter subgroup K of A. Indeed, if, for every nilpotent subgroup N of A, the Carter subgroups of N G are conjugate then A satisfies (C). Hence the Carter subgroups of A are conjugate, that contradicts the choice of A. So there exists a nilpotent subgroup N of A such that the Carter subgroups of N G are not conjugate. Let K be a Carter subgroup of N G. Clearly, then KG/G is a Carter subgroup of N G/G, i. e., it coincides with N G/G. Therefore the Carter subgroups of KG are not conjugate and KG = A1 ∈ A. So the condition A = KG in Theorems 5.2.3, 5.3.1, and 5.4.1 is not a restriction and is used only to simplify the proving. In this Section, we consider Carter subgroups in an almost simple group A with the simple socle G = F ∗ (A) ≃ PSp2n (q). We consider such groups in the separate Section since, for the groups of type PSp2n (q), the statement of Lemma 3.2.9 is not valid and we use arguments slightly different from those that we use in the proof of Theorem 5.4.1. We need the following two technical lemmas: ′

Lemma 5.2.1. Let Op (Gσ ) = G be a canonical adjoint finite group of Lie type over a field of odd characteristic p, and −1 is not a square in the base field of G. Assume that the root system Φ of G equals Cn . Let U be a maximal unipotent subgroup of G, H be a Cartan subgroup of G normalizing U , and Q be a Sylow 2-subgroup of H. Then CU (Q) = hXr | r is a long rooti. Proof. If r is a short root then there exists a root s, with < s, r >= 1. Thus, xr (t)hs (−1) = xr ((−1) t) = xr (−t) (see [5, Proposition 6.4.1]). Therefore, if x ∈ CU (Q) and xr (t) is a nontrivial multiplier in decomposition (1) of x then r is a long root. Now, if r is a long root then, for every root s, either | < s, r > | = 2 or < s, r >= 0, i. e., xr (t)hs (−1) = xr (t). Under the condition that −1 is not a square in the base field of G (i. e., in the field Fq ), we obtain that q ≡ −1 (mod 4), so, hhs (−1) | s ∈ Φi = Q.The lemma is proved. Lemma 5.2.2 [46, Theorem 2]. Let G = PSp2n (q) be a simple canonical group of Lie type, let J be a subset of the set of fundamental roots containing a long fundamental root rn , let PJ be a parabolic subgroup generated by a Borel subgroup B and by the groups Xr , with −r ∈ J, and let L be a Levi factor of PJ . Denote by S a quasisimple normal subgroup of L isomorphic to Sp2k (q) (it always exists since rn ∈ J). Then AutL (S/Z(S)) = S/Z(S). SIBERIAN ADVANCES IN MATHEMATICS

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Theorem 5.2.3. Let G be a finite adjoint group of Lie type (not necessary simple) over a field ′ of characteristic p, and let G and σ be chosen so that PSp2n (pt ) ≃ Op (Gσ ) ≤ G ≤ Gσ . Choose a subgroup A of Aut(PSp2n (pt )), with A ∩ Gσ = G. Let K be a Carter subgroup of A. Assume also that |PSp2n (pt )| 6 Cmin and A = KG. Then exactly one of the following statements holds: (1) G is defined over GF (2t ), the field automorphism ζ is in A, |ζ| = t, and up to conjugation in G, the equality K = Q ⋋ hζi holds where Q is a Sylow 2-subgroup of Gζ2′ ; (2) G ≃ PSL2 (3t ) ≃ PSp2 (3t ), the field automorphism ζ is in A, |ζ| = t is odd, and up to conjugation in G, the equality K = Q ⋋ hζi holds, where Q is a Sylow 3-subgroup of Gζ3′ ; (3) p does not divide |K ∩ G|, and K is contained in the normalizer of a Sylow 2-subgroup of A. In particular, the Carter subgroups of A are conjugate, i. e., if A1 ∈ A and F ∗ (A1 ) = Cmin then F ∗ (A1 ) 6≃ PSp2n (pt ). Proof. Assume that the theorem is not valid and A is a counter example such that |F ∗ (A)| is minimal. Note that no more than one statement of the theorem can be fulfilled since if statement (2) holds then, by Lemmas 4.3.2 and 4.3.3, for a Sylow 2-subgroup Q of A, the condition NG (S) = SCG (S) is not fulfilled, i. e., statement (3) of the theorem does not hold. Thus, if A1 is an almost simple group, with F ∗ (A1 ) being a simple group of Lie type of order less than |F ∗ (A)|, then the Carter subgroups of A1 are conjugate. In view of Theorem 1.5.6, we can assume that A 6= G. Moreover, by Theorem 3.3.5, we can assume that q is odd, i. e., that Aut(PSp2n (q)) does not contain a graph automorphism. Thus we can assume that A = hG, ζgi. Assume that K is a Carter subgroup of hG, ζgi and K does not satisfy to the statement of the theorem. Write K = hx, K ∩ Gi. If either p 6= 3 or t is even then the theorem follows from Theorem ′ 3.3.5. Thus we can assume that q = 3t and t is odd. Since |Gσ : Op (Gσ )| = 2 and the order |ζ| is odd, we can assume that the order |ζg| is also odd and so, ζ ∈ hG, ζgi, i. e., A = G ⋋ hζi. By Lemma 3.2.3, every semisimple element of odd order is conjugate to its inverse in G. Now, for every semisimple element t ∈ G, each non-Abelian composition factor of CG (t) is a simple group of Lie type (see [6]) of order less than Cmin. Therefore, for every non-Abelian composition factor S of CA (t) and every nilpotent subgroup N ≤ CA (t), the Carter subgroups of hAutN (S), Si are conjugate. It follows that CA (t) satisfies (C). Hence, by Lemma 2.4.2, |K ∩ G| = 2α · 3β for some α, β > 0. \(q) then by [48, Theorem 2] every unipotent element is conjugate to its inverse. Since If G = PSp 2n 3 is a good prime for G then [36, Theorems 1.2 and 1.4] imply that, for any element u ∈ G of order 3, all composition factors of CG (u) are simple groups of Lie type of order less than Cmin. Thus, CA (u) satisfies (C), and hence, by Lemma 2.4.2, we obtain that K ∩ G is a 2-group. By Lemmas 4.2.5 and 4.2.6 every element x ∈ A \ G of odd order, with hxi ∩ G = {e}, centralizes some Sylow 2-subgroup of G. So, K contains a Sylow 2-subgroup of G and hence, of A, i. e., K satisfies statement (3) of the theorem. Thus we can assume that G = PSp2n (q) and β > 1, i. e., a Sylow 3-subgroup O3 (K ∩ G) of K ∩ G is nontrivial. By Lemma 4.2.1, we obtain that K ∩ G is contained in some K-invariant parabolic subgroup P of G with a Levi factor L, and up to conjugation in P , a Sylow 2-subgroup O2 (K ∩ G) of K ∩ G is contained in L. Note that all non-Abelian composition factors of P are simple groups of Lie type of order e = KO3 (P )/O3 (P ) less than Cmin. So, P and each its homomorphic image satisfy (C). The group K e e P/O3 (P )i. is isomorphic to K/O3 (K ∩ G), and by Lemma 2.4.1, K is a Carter subgroup of hK, e ∩ P/O3 (P ) ≃ O2 (K ∩ G) is a 2-group and every element x ∈ hK, e P/O3 (P )i \ P/O3 (P ) of Now K odd order, with hxi ∩ P/O3 (P ) = {e}, centralizes a Sylow 2-subgroup of P/O3 (P ) ≃ L (see Lemmas 4.2.5 and 4.2.6). Therefore, O2 (K ∩ G) contains a Sylow 2-subgroup of L, in particular, contains a Sylow 2-subgroup H2 of H. Since K is nilpotent, Lemma 5.2.1 implies that O3 (K ∩ G) ≤ CU (H2 ) = hXr | r is a long root of Φ(G)+ i. Since, for every two long positive roots r and s in Φ(G)+ , we have that r + s 6∈ Φ(G), the Chevalley commutator formula [5, Theorem 5.2.2] (Lemma 1.3.1) implies that hXr | r is a long root of Φ(G)+ i is Abelian. Since ζ is a field automorphism, it normalizes each parabolic subgroup of G containing a ζ-stable Borel subgroup. Thus, for every subset J of the set of fundamental roots Π = {r1 , . . . , rn } of Φ = Φ(G), SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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the parabolic subgroup PJ is ζ-stable. Therefore we can assume that P = PJ , where J is a proper subset of the set of fundamental roots Π of Φ. Choose the numbering of fundamental roots so that rn is a long fundamental root while the remaining fundamental roots ri are short roots. If rn ∈ J then one of the components of the Levi factor, say, G1 , is isomorphic to Sp2k (q) for some k < n (note that since A 6= G then q 6= 3). By Lemma 5.2.2, we obtain that L/CL (G1 ) = AutL (G1 /Z(G1 )) = G1 /Z(G1 ). By Lemma 2.4.1, K1 = KCL (G1 )O3 (P )/CL (G1 )O3 (P ) is a Carter subgroup of (P ⋋ hζi)/CL (G1 )O3 (P ). Since |K1 ∩ P/CL (G1 )O3 (P )| is not divisible by 3 and ζ centralizes a Sylow 2subgroup of G1 /Z(G1 ) (see Lemma 4.2.5) then K1 contains a Sylow 2-subgroup of P/CL (G1 )O3 (P ) ≃ G1 /Z(G1 ) ≃ PSp2k (q). Moreover, by Lemma 4.2.5 a Sylow 2-subgroup of (P/CL (G1 )O3 (P ))ζ is a Sylow 2-subgroup of P/CL (G1 )O3 (P ). Thus K1 ∩ P/CL (G1 )O3 (P ) is a Sylow 2-subgroup of (P/CL (G1 )O3 (P ))ζ ≃ PSp2k (3). By Lemma 4.3.2, there exists an element x of odd order of PSp2k (3) that normalizes but not centralizes a Sylow 2-subgroup, a contradiction with the fact that K1 is a Carter subgroup of (P ⋋ hζi)/CL (G1 )O3 (P ). Thus we can assume that rn 6∈ J. Consider the set Jn = Π \ {rn } and the parabolic subgroup PJn . From the above arguments it follows that K ≤ PJ ⋋ hζi ≤ PJn ⋋ hζi. Now the subgroup hXr | r is a long root of Φ(G)+ i is contained in O3 (PJn ) and O3 (K ∩ G) is contained in hXr | r is a long root of Φ(G)+ i, so NG (O3 (K ∩ G)) ≤ e = KO3 (P )/O3 (P ) is a Carter subgroup O3 (PJn ) and we can assume that P = PJn . By Lemma 2.4.1, K of (P ⋋ hζi)/O3 (P ). Note that the unique non-Abelian composition factor of P ⋋ hζi is isomorphic to e = R × hζi, where R is An−1 (q) ≃ PSLn (q). By [28, Theorem 1] and [29, Theorem 4], we obtain that K a Sylow 2-subgroup of P centralized by ζ. Thus, O3 (K ∩ G) ≤ CP (R). Consider Q = O3 (K ∩ G) ∩ Pζ . Since O3 (K ∩ G) is nontrivial and K is nilpotent, then Q = O3 (K ∩ G) ∩ Pζ = Z(K) ∩ O3 (K ∩ G) is nontrivial. Therefore, NG (Q) is a proper subgroup of G and, by Lemma 4.2.1, NG (Q) is contained in a proper parabolic subgroup of G. On the other hand, K ≤ NG (Q) and P = PJn is a maximal proper parabolic subgroup of G. If NG (Q) is not contained in P then NG (Q) and K are contained in a parabolic subgroup PJ , with rn ∈ J. We proved above that rn 6∈ J, so, NG (Q) is contained in P . Prove that R × Q is a Carter subgroup of Gζ . Indeed, assume that an element x ∈ Gζ normalizes R × Q. Then x normalizes Q, so, x is in P and normalizes O3 (P ). On the other hand, x normalizes R, therefore normalizes CP (R), so x normalizes CO3 (P ) (R). Moreover, it is evident that x and ζ commute. Thus x normalizes (R × CO3 (P ) (P )) ⋋ hζi. As we noted above, K ≤ (R × CO3 (P ) (P )) ⋋ hζi and (R × CO3 (P ) (P )) ⋋ hζi is solvable. Lemma 2.4.2(a) implies that (R × CO3 (P ) (P )) ⋋ hζi coincides with its normalizer in G ⋋ hζi, so x ∈ R × CO3 (P ) (R). The group CO3 (P ) (R) ≤ hXr | r is a long root of Φ(G)+ i is Abelian, so every element of R × CO3 (P ) (R) centralizes CO3 (P ) (R) ≥ O3 (K ∩ G). Therefore x normalizes (R × CO3 (P ) (P )) ⋋ hζi = K, i. e., x ∈ K. By construction, R × Q = K ∩ Gζ , so x ∈ R × Q and ′ R × Q is a Carter subgroup of Gζ . On the other hand, O3 (Gζ ) ≃ PSp2n (3t/|ζ| ) and by induction, the t/|ζ| ) do not contain Carter subgroups of order divisible by 3. This groups PSp2n (3t/|ζ| ) and PSp\ 2n (3 final contradiction completes the proof. 5.3. Groups with triality automorphism. Theorem 5.3.1. Let G be a finite adjoint group of Lie type over a field of characteristic p, and ′ ′ let G and σ be chosen so that Op (Gσ ) ≤ G ≤ Gσ , where Op (Gσ ) is isomorphic to either D4 (q) or 3 D (q 3 ). Assume that τ is a graph automorphism of order 3 of O p′ (G) (recall that for G ≃ 3 D (q 3 ) 4 4 τ is an automorphism such that the set of its stable points is isomorphic to G2 (q)). Denote by A1 the subgroup of Aut(D4 (q)) generated by the inner-diagonal and field automorphisms, and also ′ by a graph automorphism of order 2. Let A ≤ Aut(G) be such that A 6≤ A1 (if Op (G) ≃ D4 (q)), ′ and let K be a Carter subgroup of A. Assume also that |Op (G)| 6 Cmin, G = A ∩ Gσ , and A = KG. Then one of the following statements holds: (a) G ≃ 3 D4 (q 3 ), (|A : G|, 3) = 1, q is odd and K contains a Sylow 2-subgroup of A; (b) (|A : G|, 3) = 3, q is odd, τ ∈ A, and up to conjugation by an element of G, the subgroup K contains a Sylow 2-subgroup of CA (τ ) ∈ ΓG2 (q), and τ ∈ K; SIBERIAN ADVANCES IN MATHEMATICS

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(c) (|A : G|, 3) = 3, q = 2t , |A : G| = 3t, and A = G ⋋ hτ, ϕi, where ϕ is a field automorphism of order t commuting with τ , and up to conjugation by an element of G, the subgroup K contains a Sylow 2-subgroup of CG (hτ, ϕi2′ ) ≃ G2 (2t2′ ), and τ ∈ K; ′

(d) Op (G) ≃ D4 (p3t ), p is odd, the factor group A/G is cyclic, τ 6∈ A, A = G ⋋ hζi, where, for some natural m, ζ = τ ϕm is a graph-field automorphism, and up to conjugation by an element of G, K = Q ⋋ hζi, where Q is a Sylow 2-subgroup of CG (ζ2′ ) ≃ 3 D4 (p3t/|ζ2′ | ). In particular, the Carter subgroups of A are conjugate, i. e., if A2 ∈ A and |F ∗ (A2 )| = Cmin then A2 does not satisfy to the conditions of the theorem, so, F ∗ (A2 ) 6≃ 3 D4 (q 3 ). ′

Proof. Assume that the theorem is not valid and A is a counter example such that |Op (G)| is minimal. In view of [40, Theorem 1.2(vi)], we have that every element of G is conjugate to its inverse. By [6] and [36, Theorems 1.2 and 1.4], we obtain that, for every element t ∈ G of odd prime order, all non-Abelian composition factors of CG (t) are simple groups of Lie type of order less than Cmin. Thus, CA (t) satisfies (C), and Lemma 2.4.2 implies that KG = K ∩ G is a 2-group. Now Lemma 4.2.6 implies that all cyclic groups generated by the field automorphisms of the same odd order of G, are conjugate under G. Since the centralizer of every field automorphism in G is a group of Lie type of order less than Cmin, we again use Lemma 2.4.2 and obtain the statement of the theorem by induction. Lemma 4.2.6 ′ implies also that if Op (G) ≃ D4 (q) then all cyclic groups generated by graph-field automorphisms are conjugate. Since the centralizer of each graph-field automorphism in G is a group of Lie type of order less than Cmin, we again use Lemma 2.4.2 and obtain statement (d) of the theorem by induction. Thus we can assume that A does not contain a field automorphism or a graph-field automorphism of odd order. Therefore, either G ≃ 3 D4 (q 3 ) and A/G is a 2-group or K contains an element s of order 3 such that hsi ∩ A1 = {e} (for groups 3 D4 (q 3 ), the equality hsi ∩ G = {e} holds), G ⋋ hsi = G ⋋ hτ i, and K ∩ G is a 2-group. In the first case, we obtain the statement (a) of the theorem with condition (|A : G|, 3) = 1. In the second case, there exist two nonconjugate cyclic subgroups hτ i and hxi of order 3 of A such that hτ i ∩ A1 = hxi ∩ A1 = {e} and G ⋋ hxi = G ⋋ hτ i (see [22, (9-1)]). Hence, either s = τ ∈ K or s = x ∈ K. Assume that q 6= 3t . In the first case, from the known structure of Carter subgroups in a group from the set ΓG2 (q) obtained in Theorem 3.3.5, the statement (b) or (c) of the theorem follows. In the second case, we have that K ≤ CA (x). By [22, (9-1)], CG (x) ≃ PGLε3 (q), where q ≡ ε1 (mod 3), − ε = ± and PGL+ 3 (q) = PGL3 (q), PGL3 (q) = PGU3 (q). Then K = (K ∩ G) × hy, ϕi, where ϕ is a ′ field automorphism of Op (G) of order equal to a power of 2, and y is a graph automorphism such that its order is a power of 3, and x ∈ hyi. By nilpotency of K, we obtain that yϕ = ϕy. From here it follows that CCG (ϕ) (x) = CCG (x) (ϕ). Now we have that   D (q 1/|ϕ| ), if Op′ (G) ≃ D (q), 4 4 CG (ϕ) =  3 D (q 3/|ϕ| ), if G ≃ 3 D (q 3 ). 4

4

Hence, CCG (x) (ϕ) = CCG (ϕ) (x) ≃ PGLµ3 (q 1/|ϕ| ), with q 1/|ϕ| ≡ µ1 (mod 3), where µ = ± (note that ε and µ may be different). As noted above, K ∩ G is a 2-group. On the other hand, by [29, Theorem 4], there exists an element z of order 3 centralizing a Sylow 2-subgroup of CG (x) = PGLε3 (q) and belonging to CCG (x) (ϕ) ≃ PGLµ3 (q 1/|ϕ| ). Thus z centralizes K, hence, is in K. But K ∩ G does not contain elements of odd order, therefore this second case is impossible. Assume now that q = 3t . Then CG (τ ) ≃ G2 (q) and we prove the assertion. In the second case, CG (x) ≃ SL2 (q) ⋌ U , where U is a 3-group and Z(CG (x)) ∩ U 6= {e}, a contradiction with Lemma 2.4.2.

5.4. Classification theorem. Theorem 5.4.1. Let G be a finite adjoint group of Lie type (G is not necessary simple) over ′ a field of characteristic p, and let G and σ be chosen so that Op (Gσ ) ≤ G ≤ Gσ . Assume also ′ that G 6≃ 3 D4 (q 3 ). Choose a subgroup A of Aut(Op (Gσ )), with A ∩ Gσ = G, and in the case SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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′

Op (G) = D4 (q), assume that A is contained in the subgroup A1 defined in Theorem 5.3.1. Let K be a Carter subgroup of A and assume that A = KG. Then exactly one of the following statements holds: (a) G is defined over a field of characteristic 2, A = hG, ζg, ti, where t is a 2-element, K is contained in the normalizer of a t-stable Borel subgroup of G, and K ∩ hG, ζgi satisfies to one of the statements (a)–(f) of Lemma 4.3.4; (b) G ≃ PSL2 (3t ), the field automorphism ζ is in A, |ζ| = t is odd, and up to conjugation in G, the equality K = Q ⋋ hζi holds, where Q is a Sylow 3-subgroup of Gζ3′ ; (c) A = 2 G2 (32n+1 ) ⋋ hζi, |ζ| = 2n + 1, and up to conjugation in G, the equality K = (K ∩ G) ⋋ hζi holds, and K ∩ 2 G2 (32n+1 ) = Q × P , where Q is of order 2 and |P | = 3|ζ|3 . (d) p does not divide |K ∩ G| and K contains a Sylow 2-subgroup of A. In view of Lemma 4.3.3, A satisfies (ESyl2) if and only if G satisfies (ESyl2). In particular, the Carter subgroups of A are conjugate. Remark. There exists a dichotomy for the Carter subgroups in the groups of automorphisms of finite groups of Lie type not containing a graph or a graph-field automorphism of order 3. They either are contained in the normalizer of a Borel subgroup, or the characteristic is odd and a Carter subgroup contains a Sylow 2-subgroup of the hole group. Assume that the assertion is not valid and A is a counter example to the theorem with |F ∗ (A)| minimal. Among the counter examples with |F ∗ (A)| minimal, take those for which |A| is minimal. In this case, for every almost simple group A1 such that |F ∗ (A1 )| < |F ∗ (A)|, F ∗ (A1 ) is a finite simple group of Lie type and A1 satisfies the conditions of Theorem 5.4.1, the Carter subgroups are conjugate. Indeed, note that no more than one statement of the theorem can be fulfilled since if either statement (b) or statement (c) of the theorem holds then the condition NA (Q) = QCA (Q) for a Sylow 2-subgroup Q of A is not valid, i. e., the statement (d) of the theorem does not hold (the fact that other statements cannot hold simultaneously is evident). Thus, the Carter subgroups of A1 are conjugate. Note also that from this fact we immediately obtain the inequality |F ∗ (A)| 6 Cmin. Indeed, if A2 ∈ A and F ∗ (A2 ) = Cmin then either A2 satisfies to the condition of Theorem 5.3.1 or A2 satisfies conditions of Theorem 5.4.1. As noted in Theorem 5.3.1, the first case is impossible. The second case, as we just noted, is possible only if |F ∗ (A)| 6 |F ∗ (A2 )| = Cmin (since A is a counterexample to the statement of the theorem with |F ∗ (A)| is minimal). We will prove the theorem in the following way. If F ∗ (A) ≃ PSp2n (q) then the theorem follows from Theorem 5.2.3. If A = G then the theorem follows from [15–17, 33, 41] and the results from Section 3 of the present paper. Thus we can assume that A/(A ∩ G) is nontrivial. Let K be a Carter subgroup of A. We will prove first that if p divides |K ∩ G| then one of the statements (a)–(c) of the theorem holds. Then we will prove that if p does not divide |K ∩ G| then K contains a Sylow 2-subgroup of A. Since both of these steps are quite complicated, we divide them into two Subsections. Note that, in view of [6], for every semisimple element t ∈ G, all non-Abelian composition factors of CG (t), so, of CA (t), are simple groups of Lie type of order less than |F ∗ (A)|, and hence, of order less than Cmin. Therefore, CA (t) satisfies (C). In the sequel, to apply Lemmas 2.4.1 and 2.4.2, we will use this fact without any references. 5.5. Carter subgroups of order divisible by characteristic. Denote K ∩ G by KG . For every group A satisfying conditions of Theorem 5.4.1, the factor group A/G is Abelian and, for some natural t, is isomorphic to a subgroup of Z2 × Zt , where Zt denotes a cyclic group of order t. If the factor group ′ A/G is not cyclic then the group Op (G) is split and A contains an element τ a, where τ is a graph ′ automorphism of Op (G) and a ∈ Gσ . Then every semisimple element of odd order is conjugate to its inverse in A (see Lemma 4.2.7). By Lemma 2.4.2, we obtain that |KG | is divisible only by 2 and p. If p = 2 then we obtain that KG is a 2-group. It is contained in a proper K-invariant parabolic subgroup P of G, and by Lemma 2.4.1,TKO2 (P )/O2 (P ) is a Carter subgroup of KP/O2 (P ). Since KG ≤ O2 (P ), then (KO2 (P )/O2 (P )) (P/O2 (P )) = {e}. Hence, P is a Borel subgroup of G, otherwise we have CP/O2 (P ) (KO2 (P )/O2 (P )) 6= {e}, a contradiction with the fact that KO2 (P )/O2 (P ) is a Carter subgroup of KP/O2 (P ). Thus, P is a Borel subgroup and the statement of the theorem follows from SIBERIAN ADVANCES IN MATHEMATICS

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Lemma 4.3.4. Now, if p 6= 2 then again KG is contained in a proper parabolic subgroup P of G such that Op (KG ) ≤ Op (P ) and O2 (KG ) ≤ L. Then Lemmas 4.2.5 and 4.2.6 imply that H2 ≤ O2 (K ∩ G) ≤ K. Now Lemma 3.2.9 implies that Op (KG ) ≤ CU (H2 ) = {e}. Therefore, K ∩ G is a 2-group. By Lemmas 4.2.5 and 4.2.6, every element x ∈ A \ G of odd order such that hxi ∩ G = {e} centralizes some Sylow 2-subgroup of G. Hence, K contains a Sylow 2-subgroup of A, i. e., K satisfies statement (d) of the theorem. Therefore, A/G is cyclic and we can assume that A = hG, ζgi ∈ ΓG. Recall that we argue under the conditions of Theorem 5.4.1, A = hG, ζgi is supposed to be a counter ′ example to the theorem with |Op (G)| and |A| minimal, and K is a Carter subgroup of hG, ζgi such ′ that p divides |KG |. We have that K = hζ k g, KG i. Since |Op (G)| 6 Cmin, Lemma 2.4.1 implies that KG/G is a Carter subgroup of hG, ζgi/G. Therefore, |ζ k | = |ζ| and we can assume that k = 1 and K = hKG , ζgi. ¯ In view of Lemma 4.2.1, there exists a proper σ- and ζg-invariant parabolic subgroup P of G such ζ¯

that Op (KG ) ≤ Ru (P ) and KG ≤ P . In particular, P and P are conjugate in G. Let Φ be the root system of G and Π be a set of fundamental roots of Φ. In view of [5, Proposition 8.3.1], P is conjugate to some P J = B · N J · B, where J is a subset of Π and N J is a complete preimage of WJ in N under the ζ¯ γ ¯ε ¯ hence, P = P (recall that ζ¯ = γ¯ ε ϕk by natural homomorphism N /T → W . Now, P J is ϕ-invariant, J

J

definition). Consider the symmetry ρ of the Dynkin diagram of Φ corresponding to γ¯ . Let J be the image ζ¯

γ ¯

ζ¯

of J under ρ. Clearly, P J = P J . Since P and P are conjugate in G, we obtain that P J and P J are ¯ conjugate in G. By [5, Theorem 8.3.3], it follows that either ε = 0, or J = J; i. e., P J is ζ-invariant. ¯

y ¯

y¯ ¯ P iy¯ = h(ζg) ¯ y¯ , P J i and P (ζg) = P J . It follows Now we have that P = P J for some y¯ ∈ G. So hζg, J ¯ y¯ = y¯−1 ζg ¯ y¯ = ζ¯ ζ¯−1 y¯−1 ζg ¯ y¯ = ζ¯ · h, (ζg) −1 ¯ ¯ y¯ ∈ G. Since P ζ = P J = P h , we obtain that h ∈ N (P J ). By [5, Theowhere h = ζ¯−1 y¯−1 ζg J J G ¯ P iy = hζ, ¯ P J i. Further, both P and P J are σ-invariant. Hence, rem 8.3.3], NG (P J ) = P J , thus, hζg, y¯σ(¯ y −1 ) ∈ NG (P ) = P . Therefore, by the Lang-Steinberg Theorem (Lemma 1.5.3), we can assume ′ that y¯ = σ(¯ y ), i. e., y¯ ∈ Gσ . Since Gσ = T σ · Op (Gσ ) and T ≤ P J , then we can assume that y¯ ∈ ′ ¯ P J i = P J ⋋ hζi ¯ and Op (Gσ ). Thus, up to conjugation in G, we can assume that K ≤ hζ,

K ≤ h(P J ∩ G), ζgi = hPJ , ζgi, ¯ in particular, g ∈ (P J )σ . Further, if LJ = hT , X r | r ∈ J ∪ −Ji then LJ is a σ- and ζ-invariant Levi g factor of P J and LJ = LJ ∩ G is a ζ-invariant Levi factor of PJ . Then LJ is a ζg-stable factor Levi of PJ . Since all Levi factors are conjugate under Op (PJ ), we can assume that LJ is a ζg-stable Levi factor. Lemma 2.4.1 implies that KOp (PJ )/Op (PJ ) = X is a Carter subgroup of hPJ , ζgi/Op (PJ ), and e KZ(LJ )Op (PJ )/Z(LJ )Op (PJ ) = X

is a Carter subgroup of hPJ , ζgi/Z(LJ )Op (PJ ). Recall that K = hζg, KG i. Hence, if v and v˜ are the images of g under the natural homomorphisms ω : hPJ , ζgi → hLJ , ζgi ≃ hPJ , ζgi/Op (PJ ), ω ˜ : hPJ , ζgi → hPJ , ζgi/Z(LJ )Op (PJ ) ≃ hLJ , ζgi/Z(LJ ), ω i and X e = hζ v˜, K ω˜ i. Note that Op (P ) and Z(LJ ) are characteristic subgroups of then X = hζv, KG G e= P and LJ respectively. Hence we may consider ζ as an automorphism of LJ ≃ P/Op (P ) and L LJ /Z(LJ ). Note also that all non-Abelian composition factors of P are simple groups of Lie type of

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e ζgi, hL, ζgi, order less than Cmin, hence, hP, ζgi satisfies (C). Thus we can apply Lemma 2.4.1 to hL, and hP, ζgi. If PJ is a Borel subgroup of G then the statement of the theorem follows from Lemma 4.3.4. So we can assume that LJ 6= Z(LJ ), i. e., that PJ is not a Borel subgroup of G. Then LJ = H(G1 ◦ . . . ◦ Gk ), where Gi are subsystem subgroups of G, k > 1, and H is a Cartan subgroup of G. Let ζg = (ζ2 g2 ) · (ζ2′ g2′ ) be the product of 2- and 2′ - parts of ζg (with g2 , g2′ ∈ (P J )ζ ). Now, for some k, ζ2′ = ϕk is a field automorphism (recall that we do not consider the triality automorphism) and it normalizes each Gi since ϕ normalizes each Gi . Moreover, in view of Lemma 4.2.5, we have that ζ2′ centralizes a Sylow 2subgroup of H. In particular, it centralizes a Sylow 2-subgroup of Z(LJ ) ≤ H. Therefore, every element of odd order of hLJ , ζ2′ v2′ i centralizes a Sylow 2-subgroup of Z(LJ ) (here v2′ is the image of g2′ under ω). e = (PG1 × . . . × PGk )H, e where H e = H ω and PG1 , . . . , PGk are canonical finite groups Now L of Lie type with trivial center. Set Mi = CLe (PGi ). Clearly, Mi = (PG1 × . . . × PGi−1 × PGi+1 × e i and by πi the corresponding natural . . . × PGk )CHe (PGi ). Denote by Li the factor group L/M di. homomorphism. Then Li is a finite group of Lie type and PGi ≤ Li ≤ PG Set Mi,j = CLe (PGi × PGj ). Then Mi,j = (PG1 × . . . × PGi−1 × PGi+1 × . . . × PGj−1 × PGj+1 × . . . × PGk )CHe (PGi × PGj ). e → L/M e i,j . If Mi (respectively, Mi,j ) is ζDenote by πi,j the corresponding natural homomorphism L e ζ v˜i and we denote by πi (respectively, πi,j ) the invariant then Mi (respectively, Mi,j ) is normal in hL, e e e ζ v˜i → hL, e ζ v˜i/Mi,j ). natural homomorphism πi : hL, ζ v˜i → hL, ζ v˜i/Mi (πi,j : hL, Now, consider ζ. Since ζ 2 is a field automorphism, there are only two cases: Either ζ normalizes PGi or ζ 2 normalizes PGi , and PGζi = PGj for some j 6= i. Consider these two cases separately. e πi = Ki is a Carter Let ζ normalize PGi . Then ζ normalizes Mi and Lemma 2.4.1 implies that X π π i i subgroup of hLi , (ζ v˜) i. Since hLi , (ζ v˜) i is a semilinear group of Lie type satisfying the conditions of Theorem 5.4.1 (by definition, ζ 2 is a field automorphism, so we are not in the conditions of Theorem 5.3.1), |Li | < |G|, and p does not divide |Ki |, we have that Ki contains a Sylow 2-subgroup Qi of hLi , (ζ v˜)πi i (in particular, p 6= 2), and by Lemma 2.4.3, the group hLi , (ζ v˜)πi i satisfies (ESyl2). e ζ v˜i. We want to show that Let ζ 2 normalizes PGi and PGζi = PGj . Then Mi,j is normal in hL, π e i,j e e πi,j hL, ζ v˜i satisfies (ESyl2). Since Mi,j is a normal subgroup of hL, ζ v˜i then, by Lemma 2.4.1, (X) e ζ v˜iπi,j . Consider the subgroup is a Carter subgroup of hL, e πi,j i h(PGi )πi,j × (PGj )πi,j , X

e ζ v˜iπi,j (note that (PGi )πi,j ≃ PGi and (PGj )πi,j ≃ PGj , and in this Section, we will identify of hL, these groups for brevity). Now we are in the conditions of Lemma 2.2.3, namely, we have a finite e = (X) e πi,j (PGi × PGj ), where PGi ≃ PGj has a trivial center. Then Aut e πi,j (PGi ) ≃ group G (X) AutXe (PGi ) is a Carter subgroup of AutGe (PGi ). Now PGi is a canonical finite group of Lie type and PGi ≤ AutGe (PGi ) ≤ Aut(PGi ), i. e., AutGe (PGi ) satisfies the conditions of Theorem 5.4.1 (by construction ζ 2 is a field automorphism e πi,j ∩ (PGi × PGj ) is not divisible by the and so we are not in the conditions of Theorem 5.3.1) and (X) characteristic. By induction, Aut(X) e (PGi ) (in particue πi,j (PGi ) contains a Sylow 2-subgroup of AutG lar, p 6= 2). The same arguments show that AutXe (PGj ) contains a Sylow 2-subgroup of AutGe (PGj ). Therefore, AutGe (PGi ) and AutGe (PGj ) satisfy (ESyl2). Since AutGe (PGi ) ≤ AuthL,ζ e v˜iπi,j (PGi ) and AutGe (PGj ) ≤ AuthL,ζ e v˜iπi,j (PGj ), Lemmas 4.3.1 and 4.3.3 imply that the groups of induced automorphisms AuthL,ζ e v˜iπi,j (PGi ) and AuthL,ζ e v˜iπi,j (PGj ) satisfy (ESyl2). Consider NhL,ζ e v˜iπi,j (PGi ) and NhL,ζ e v˜iπi,j (PGj ). Since e ζ v˜iπi,j : N e πi,j (PGi )| = |hL, e ζ v˜iπi,j : N e πi,j (PGj )| = 2, |hL, hL,ζ v˜i hL,ζ v˜i

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e ζ v˜iπi,j , the equality of cosets hN e πi,j (PGi ) = it is easy to see that, for every element h of hL, hL,ζ v˜i hNhL,ζ (PG ) holds, it follows that N (PG ) = N (PG ). By construction, π π π j i j e v˜i i,j e v˜i i,j e v˜i i,j hL,ζ hL,ζ ChL,ζ e v˜iπi,j (PGi ) ∩ ChL,ζ e v˜iπi,j (PGj ) = {e}, so Lemma 2.4.5 (with ChL,ζ e v˜iπi,j (PGi ) and ChL,ζ e v˜iπi,j (PGj ) as normal subgroups) implies that the nore ζ v˜iπi,j : N e πi,j (PG1 )| = 2. Thus Lemma malizer N e πi,j (PGi ) satisfies (ESyl2). Further, |hL, hL,ζ v˜i

hL,ζ v˜i

e ζ v˜iπi,j satisfies (ESyl2). 2.4.6 implies that hL,

e 6= {e}, then, as noted above, p 6= 2. Let Now we will show that hLJ , ζvi satisfies (ESyl2). Since L Q be a Sylow 2-subgroup of hLJ , ζvi. Consider an element x ∈ NhLJ ,ζvi (Q) of odd order. We need to prove that x centralizes Q. As noted above, every element of odd order of hLJ , ζvi centralizes Q ∩ Z(LJ ). e = Qω˜ ≃ Q/(Q ∩ Z(LJ )) then x centralizes Q. Now either Mi is normal Hence, if x ˜ = xω˜ centralizes Q T e ζ v˜i or Mi,j is normal in hL, e ζ v˜i and (∩i Mi ) (∩i,j Mi,j ) = {e}. Moreover, as proved above, xπi in hL, e i /Mi and xπi,j centralizes QM e i,j /Mi,j . By Lemma 2.4.5 (with normal subgroup Mi and centralizes QM e Mi,j ), we obtain that x ˜ centralizes Q. Thus hLJ , ζvi satisfies (ESyl2), and by Lemma 2.4.3, there exists a Carter subgroup F of hLJ , ζvi containing Q. Since hLJ , ζvi satisfies (C), Theorem 2.1.4 implies that X = K ω and F are conjugate, i. e., X contains a Sylow 2-subgroup of hLJ , ζvi, and up to conjugation in hPJ , ζvi, K contains a Sylow 2-subgroup of hPJ , ζvi. In particular, a Sylow 2-subgroup Q1 of a Cartan subgroup H is in K and Q1 centralizes K ∩ Op (PJ ) 6= {e}, a contradiction with Lemma 3.2.9. 5.6. Carter subgroups of order not divisible by characteristic. We are again in the conditions of Theorem 5.4.1. As noted in the previous Section, for every group A satisfying the conditions of Theorem 5.4.1, the factor group A/G is Abelian and, for some natural t is isomorphic to a subgroup of Z2 × Zt . If ′ the factor group A/G is not cyclic then Op (G) is split and A contains an element τ a, where τ is a graph ′ automorphism of Op (G) and a ∈ Gσ . Thus, if A/G is not cyclic or Φ(G) 6= An , D2n+1 , E6 then, by Lemmas 3.2.3 and 4.2.7, every semisimple element of G is conjugate to its inverse. By Lemma 2.4.2, we obtain that KG = K ∩ G is a 2-group. In the conditions of Theorem 5.4.1, the group A/G is Abelian and if A1 is a Hall 2′ -subgroup of A/G then A1 is cyclic. Let x be the preimage of the generating element of A1 taken in K. Then hxi ∩ G ≤ hxi ∩ Gσ ≤ K ∩ Gσ = K ∩ (A ∩ Gσ ) = K ∩ G. As noted above, K ∩ G is a 2-group, hence, hxi ∩ Gσ = {e}. By Lemma 4.2.6, the element x under Gσ is conjugate to a field automorphism of odd order, and by Lemma 4.2.5, the element x centralizes a Sylow 2-subgroup of G (in particular, p 6= 2) and, since A/G is Abelian, Lemma 2.4.5 implies that K contains a Sylow 2-subgroup of A. Thus the statement of Theorem 5.4.1 holds in this case. So we can assume that A = hG, ζgi is a semilinear group of Lie type, K = hζ k g, KG i is a Carter subgroup of A, and Φ(G) ∈ {An , D2n+1 , E6 }. As in the previous Section, we can assume that k = 1. Since Gζ is nontrivial, then the centralizer CG (ζg) is nontrivial, so, KG is also nontrivial. Therefore, Z(K) ∩ KG is nontrivial. Consider an element x ∈ Z(K) ∩ KG of prime order. Then K ∈ CA (x) = hζg, CG (x)i. Now CG (x)0 = C is a connected σstable reductive subgroup of maximal rank of G. Moreover, C is a characteristic subgroup of CG (x) and CG (x)/C is isomorphic to a subgroup of ∆ (see [26, Proposition 2.10]). Thus K is contained in hK, Ci, where C = C ∩ G. Moreover, by Lemma 4.1.1, the subgroup C = C ∩ G = T (G1 ◦ . . . ◦ Gm ) is normal in CA (x) and KG C/C is isomorphic to a subgroup of ∆. Assume that |KG | is not divisible by 2. ¯ If m = 0 then C = T = Z(C) is a maximal torus. So, T is ζg-stable. In view of Lemma 4.2.4, we obtain that NA (CA (x)) 6= CA (x). This is a contradiction with Lemma 2.4.2 since CA (x) is solvable in this case. If m > 1 then Z(C) and G1 ◦ . . . ◦ Gm are normal subgroups of hK, Ci. Hence we may consider e = hK, G1 ◦ . . . ◦ Gm ◦ Z(C)i/Z(C) ≤ hK, Ci/Z(C). Then G e = K(PG e e G 1 × . . . × PGm ), where K = e e KZ(C)/Z(C) is a Carter subgroup of G (see Lemma 2.4.1) and Z(PGi ) is trivial. Further, K acts by conjugation on {PG1 , . . . , PGm } and without lost of generality we can assume that {PG1 , . . . , PGm } SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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e is a K-orbit. Thus we are in the condition of Lemma 2.2.3 and AutKe (PG1 ) is a Carter subgroup of e ∩ PG1 × . . . × PGm | is not divisible by the characteristic. By induction, we AutGe (PG1 ). Moreover, |K have that either AutKe (PG1 ) contains a Sylow 2-subgroup of AutGe (PG1 ) or AutGe (PG1 ) satisfies the conditions of Theorem 5.3.1 and AutGe (PG1 ) ∩ PG1 is a nontrivial 2-group, in particular, p is odd. In any case, |K ∩ G| is divisible by 2 that contradicts our assumption. Therefore the order |KG | is even and we can assume that x ∈ Z(K) ∩ KG is an involution. Write ζg = ζ2 g1 · ζ2′ g2 , where ζ2 g1 is the 2-part and ζ2′ g2 is the 2′ -part of ζg. By Lemma 4.2.5, the element ζ2′ centralizes a Sylow 2-subgroup QG of G, so we can assume that the order of g2 is odd. Up to conjugation in G, we can assume that ζ2′ centralizes a Sylow 2-subgroup of KG . In particular, ζ2′ centralizes x. Let Q be a Sylow 2-subgroup of CG (x). Then there exists y ∈ G such that Qy ≤ QG . Substituting the subgroup K by its conjugate K y , we can assume that ζ2′ centralizes a Sylow 2subgroup of CG (x). Since ζ2′ g2 centralizes x, we obtain that g2 ∈ CGσ (x). Moreover, by Lemma 3.2.1, it follows that g2 ∈ CG (x)0 . In particular, g2 normalizes each Gi and centralizes Z(C) and Z(CG (x)). Note that ζ2′ normalizes each Gi and centralizes a Sylow 2-subgroup of Z(CG (x)) (recall that ζ2′ centralizes a Sylow 2-subgroup of CG (x)). Indeed, ζ2′ normalizes C, hence, normalizes the characteris′ tic subgroups Op (C) = G1 ◦ . . . ◦ Gm and Z(C) of C. So we may consider the induced automorphism ζ2′ of ′

′

Op (C)/(Z(C) ∩ Op (C) = PG1 × . . . × PGm . Since each PGi has the trivial center and cannot be represented as a direct product of proper subgroups, the corollary of the Krull–Remak–Schmidt Theorem [34, 3.3.10] implies that ζ2′ permutes different PGi . Since ζ2′ centralizes a Sylow 2-subgroup of CG (x) and C E CG (x), then ζ2′ centralizes a Sylow 2subgroup of C, hence, centralizes a Sylow 2-subgroup Q1 × . . . × Qm of PG1 × . . . × PGm , where Qi is a Sylow 2-subgroup of PGi . If ζ2′ would induce a nontrivial permutation on the set {PG1 , . . . , PGm } then it would induce a nontrivial permutation on {Q1 , . . . , Qm }. Since each Qi is nontrivial, this is impossible. Thus every element of odd order of hK, Ci centralizes a Sylow 2-subgroup of Z(C) and normalizes each Gi . If Φ(G) = E6 then, by Lemma 3.2.1, the centralizer of every involution of G in G is connected. By Lemma 4.2.2, every involution of G is contained in a maximal torus T such that N (G, T )/T ≃ W , where W is a Weyl group of G. C is known to be generated by the torus T and T -root subgroups. Write C = T (G1 ◦ . . . ◦ Gk ). Since T σ either is obtained from a maximal split torus H by twisting with an element w0 of order 2 or is equal to H, and each field automorphism acts trivially on the factor group NG (H)/H, then ζ¯2′ normalizes every subgroup Gi . So, if Φ(Gi ) = D4 then ζ¯2′ induces a field (but not a graph or a graph-field) automorphism of Gi . Moreover, since σ acts trivially on the factor group NG (T )/T (see Lemma 4.2.5), then [6, Proposition 6] implies that σ normalizes each Gi . Therefore, none of Gi is isomorphic to 3 D4 (q 3 ). If Φ(G) coincides with An or Dn then [7, Propositions 7, 8, 10] imply that none of Gi is isomorphic to 3 D4 (q 3 ). Therefore, in any case, none of Gi is isomorphic to 3 D4 (q 3 ). Moreover, Lemma 3.2.1 implies that |KG : (KG ∩ C)| divides |CG (x)/CG (x)0 | and CG (x)/CG (x)0 is a 2-group. In [7], it is proven that if a root system Φ has the type Dn and Ψ is its subsystem of type D4 then none element from NW (Φ) (W (Ψ)) induces a symmetry of order 3 of the Dynkin diagram of Ψ. Since ζ 2 is a field automorphism, the lack of the symmetry of order 3 together with [6, Proposition 6] implies that, for each Gi , the automorphism ζ2′ is field (but not graph or graph-field). Therefore the group of induced automorphisms hAutKe (PGi ), PGi i satisfies the conditions of Theorem 5.4.1 for all i. e = K(PG e Now, consider G 1 × . . . × PGm ) ≤ hK, Ci/Z(C) (the case m = 0 is not excluded), where e e (see Lemma 2.4.1) and, for all i, Z(PGi ) = {e}. By K = KZ(C)/Z(C) is a Carter subgroup of G Lemma 2.2.3, we have that AutKe (PG1 ) is a Carter subgroup of AutGe (PG1 ). Since PG1 is a finite group of Lie type satisfying Theorem 5.4.1, by induction we obtain that AutGe (PG1 ) satisfies (ESyl2). Similarly we have that AutGe (PGi ) satisfies (ESyl2) for all i. Since AuthK,Ci/Z(C) (PGi ) ≥ AutGe (PGi ), SIBERIAN ADVANCES IN MATHEMATICS

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Lemmas 4.3.1 and 4.3.3 imply that AuthK,Ci/Z(C) (PGi ) satisfies (ESyl2). Since ChK,Ci/Z(C) (PG1 × . . . × PGm ) = {e}, Lemma 2.4.5 in which we consider the normal subgroups ChK,Ci/Z(C) (PG1 ) ∩ NhK,Ci/Z(C) (PG1 ), . . . , ChK,Ci/Z(C) (PGm ) ∩ NhK,Ci/Z(C) (PG1 ), implies that NhK,Ci/Z(C) (PG1 ) satisfies (ESyl2). Further, |hK, Ci/Z(C) : NhK,Ci/Z(C) (PG1 )| = 2t , and each element of odd order of hK, Ci/Z(C) normalizes PG1 . Thus, by Lemma 2.4.6, we obtain that the factor group hK, Ci/Z(C) satisfies (ESyl2) and, by Lemma 2.4.5, hK, Ci satisfies (ESyl2). Since |PGi | < Cmin, then hK, Ci satisfy (C). By Lemma 2.4.3, we obtain that there exists a Carter subgroup F of hK, Ci containing a Sylow 2-subgroup of hK, Ci. By Theorem 2.1.4, the subgroups F and K are conjugate in hK, Ci, thus K contains a Sylow 2-subgroup Q of hK, Ci. Since |CG (x) : C| is a power of 2 and hK, Ci normalizes CG (x), we obtain that |hK, CG (x)i : hK, Ci| is a power of 2. Moreover, by construction, each element of odd order of hK, CG (x)i is in hK, Ci. Thus, by Lemma 2.4.6, hK, CG (x)i satisfies (ESyl2) and K contains a Sylow 2-subgroup Q of hK, CG (x)i. Let ΓQ be a Sylow 2-subgroup of hG, ζgi containing Q and t ∈ Z(ΓQ) ∩ G. Then t ∈ CG (x), hence, t ∈ Z(Q) and t ∈ Z(K). Thus we can substitute x by t in the arguments above and obtain that Q = ΓQ, i. e., K contains a Sylow 2-subgroup of hG, ζgi. It completes the proof of Theorem 5.4.1. 5.7. Carter subgroups of finite groups are conjugate. Before we formulate the main theorem, note the following corollary of Theorem 5.4.1. Corollary 5.7.1. Cmin = ∞, i. e., A = ∅. Proof. Indeed, let A 6= ∅ and A ∈ A be such that the equality |F ∗ (A)| = Cmin holds. Since ′ ∗ F (A) = Op (Gσ ) for an adjoint simple connected linear algebraic group G and a Frobenius map σ, denote the intersection A ∩ Gσ by G. As noted in the beginning of Subsection 5.1, we can assume that A = KF ∗ (A) = KG. Therefore A satisfies either the conditions of Theorem 5.3.1 or the conditions of Theorem 5.4.1. In both the cases, we proved that the Carter subgroups of A are conjugate, that contradicts the choice of A. In order to state the main theorem without the classification of finite simple groups, we give the following definition. A finite group is said to be a K-group if all its non-Abelian composition factors are known simple groups. Theorem 5.7.2 (Main Theorem). Let G be a finite K-group. Then the Carter subgroups of G are conjugate. Proof. By Theorems 1.5.6, 3.3.5, 5.2.3, 5.3.1, and 5.4.1 of the present paper, and also by [15– 17, 33, 41], we obtain that, for each known simple group S and each nilpotent subgroup N of a group of its automorphisms, the Carter subgroups of hN, Si are conjugate. So, G satisfies (C). Hence, by Theorem 2.1.4, the Carter subgroups of G are conjugate. From Lemma 2.4.1 and Main Theorem 5.7.2, it follows that a homomorphic image of a Carter subgroup is a Carter subgroup as well. Theorem 5.7.3. Let G be a finite K-group, H a Carter subgroup of G, and N a normal subgroup of G. Then HN/N is a Carter subgroup of G/N . 6. EXISTENCE CRITERION 6.1. Brief review of the results. In this Section, we will obtain a criterion of the existence of Carter subgroups in a finite group in terms of its normal series. Note that there exist finite groups without Carter subgroups, a minimal counter example is Alt5 . We will construct an example showing that an essential improvement of the criterion is impossible. At the end of this Section, for convenience of the reader, we assemble the classification of Carter subgroups in finite almost simple groups that is obtained in the present paper. Recall that, in view of Theorem 5.7.2, in every almost simple group with a known simple socle, the Carter subgroups are conjugate. Thus, modulo the classification of finite simple groups, in every finite SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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group, the Carter subgroups are conjugate. In this Section, by a finite group we always mean a finite group satisfying (C). Thus the results of this Section do not depend on the classification of finite simple groups. Definition 6.1.1. Let G = G0 ≥ G1 ≥ . . . ≥ Gn = {e} be a chief series of G (recall that G is assumed to satisfy (C)). Then Gi /Gi+1 = Ti,1 × . . . × Ti,ki , where Ti,1 ≃ . . . ≃ Ti,ki ≃ Ti and Ti is a simple group. For i > 1, denote by K i a Carter subgroup of G/Gi (if it exists) and by Ki its complete preimage in G/Gi+1 . If i = 0 then K 0 = {e} and K0 = G/G1 (note that K 0 always exists). A finite group G is said to satisfy (E) if, for each i, j, either K i does not exists or AutKi (Ti,j ) contains a Carter subgroup. By Theorem 6.2.2 and Theorem 5.7.3, it follows that if a finite group satisfies (E) then, for every i, the subgroup K i exists, so the first part of the condition (E) is never satisfied. Recall that, by Theorem 5.7.3, a homomorphic image of a Carter subgroup is a Carter subgroup as well. In the sequel, we will use this fact. 6.2. Criterion. Below we will need an additional information on a structure of Carter subgroups in groups of special type. Let A′ be a group with a normal subgroup T ′ . Consider the direct product A1 × . . . × Ak , where A1 ≃ . . . ≃ Ak ≃ A′ , and its normal subgroup T = T1 × . . . × Tk , where T1 ≃ . . . ≃ Tk ≃ T ′ . Consider the symmetric group Symk acting on A1 × . . . × Ak by the rule Asi = Ais for all s ∈ S. Define X to be equal to the semidirect product (A1 × . . . × Ak ) ⋋ Symk (the permutation wreath product of A′ and Symk ). Denote by A the direct product A1 × . . . × Ak and by πi the projection πi : A → Ai . In these notation, the following statement holds. Lemma 6.2.1. Let G be a subgroup of X such that T E G, G/(G ∩ T ) is nilpotent and (G ∩ A)πi = Ai . Assume also that A is solvable. Let K be a Carter subgroup of G. Then (K ∩ A)πi is a Carter subgroup of Ai . Proof. Assume that the statement is not valid and let G be a counter example of minimal order with k minimal. Then S = G/(G ∩ A) is transitive and primitive. Indeed, if S is not transitive then S ≤ Symk1 × Symk−k1 , hence, G ≤ G1 × G2 . If we denote by ψi : G → Gi the natural homomorphism then Gψi = Gi satisfies the conditions of the lemma and K ψi = Ki is a Carter subgroup of Gi . Clearly, (G ∩ A)πj = (Gi ∩ Aψi )πj , where i = 1 if j ∈ {1, . . . , k1 } and i = 2 if j ∈ {k1 + 1, . . . , k}, i. e., the following diagrams are commutative: G ∩ AK

πj

KK KKψ1 KK KK K%%

// Aj , :: u πj uuu u uu uu

G ∩ AK

πj

KK KKψ2 KK KK K%%

G1 ∩ Aψ1

// Aj . :: u πj uuu u uu uu

G2 ∩ Aψ2

Thus we obtain the statement by induction. If S is transitive but is not primitive, let Ω1 = {T1 , . . . , Tm }, Ω2 = {Tm+1 , . . . , T2m }, . . . , Ωl = {T(l−1)m+1 , . . . , Tlm } be a system of imprimitivity. Then it contains a nontrivial nontransitive normal subgroup F ′ ≤ Symm × . . . × Symm , {z } | l times

where k = m · l. Consider a complete preimage F of F ′ in X. Then G ∩ F ≤ F1 × . . . × Fl . Denote by ψi : F → Fi a natural projection. Then (G ∩ F )ψi = Fi . Note that all Fi satisfy the conditions of the lemma and if we define Ti′ = T(i−1)m+1 × . . . × Tim then G satisfies the conditions of the lemma, with T ′ = T1′ × . . . × Tl′ and A′ = F . By induction, we have that (K ∩ F )ψi is a Carter subgroup of π Fi and if j ∈ {m · (i − 1) + 1, . . . m · i} then (K ∩ F )ψi ∩ Aψi j is a Carter subgroup of Aj . Since π (G ∩ A)πj = (K ∩ F )ψi ∩ Aψi j (for a suitable i), we get the statement by induction. Let Y ′ be a minimal normal subgroup of G contained in T (if Y ′ is trivial then T is trivial and we have nothing to prove since G is nilpotent in this case). Thus Y ′ is a normal elementary Abelian p-group. Let SIBERIAN ADVANCES IN MATHEMATICS

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Yi = (Y ′ )πi . Then Y = Y1 × . . . × Yk is a nontrivial normal subgroup of G (Y is a subgroup of G since T ≤ G). Let π ¯i : (G ∩ A) → Ai /Yi = Ai be a projection corresponding to πi . Denote by K = KY /Y the corresponding Carter subgroup of G = G/Y . Then G satisfies the conditions of the lemma. By induction, (K ∩ A)π¯i is a Carter subgroup of Ai . Let K1 be a complete preimage of K in G and let Q be a Hall p′ -subgroup of K1 . Then (Q ∩ A)πi is a Hall p′ -subgroup of (K1 ∩ A)πi . In view of the proof of [27, Theorem 20.1.4], we obtain that K = NK1 (Q) is a Carter subgroup of G and (NK1 ∩A (Q ∩ A))πi is a Carter subgroup of Ai . Thus we need to show that (NK1 ∩A (Q ∩ A))πi = (NK1 ∩S (Q))πi . By induction, the equality (NK∩A (A ∩ Q))π¯i = (NK∩G (Q))π¯i holds. Thus we need to prove that (NY (Q ∩ A))πi = (NY (Q))πi . Note also that (NY (Q ∩ A))πi ≤ NYi ((Q ∩ A)πi ). Since S is transitive and primitive subgroup of Symk , then k = r is a prime and S = hsi is cyclic. If r = p then Q ∩ A = Q and we have nothing to prove. Otherwise, let h be an r-element of K generating S modulo K ∩ A. Clearly, Q = (Q ∩ A)hhi. Let t ∈ Yi be an element of NYi ((Q ∩ A)πi ). Then r−1 r−1 (t · th · . . . · th ) ∈ NY (Q) and tπi = (t · th · . . . · th )πi . Hence, (NY (Q ∩ A))πi ≤ NYi ((Q ∩ A)πi ) ≤ (NY (Q))πi ≤ (NY (Q ∩ A))πi . Theorem 6.2.2. Let G be a finite group. Then G contains a Carter subgroup if and only if G satisfies (E). Proof. We first prove the part “only if”. Let H be a minimal normal subgroup of G. Then H = T1 × . . . × Tk , where T1 ≃ . . . ≃ Tk ≃ T are simple groups. If H is elementary Abelian (i. e., T is cyclic of prime order) then Aut(T ) is solvable and contains a Carter subgroup. Assume that T is a non-Abelian simple group. Clearly, K is a Carter subgroup of KH. By Lemma 2.2.3, we obtain that AutKH (Ti ) contains a Carter subgroup for all i. Induction on the order of the group completes the proof of the necessity. Now we prove the part “if”. Again assume by contradiction that G is a counter example of minimal order, i. e., that G does not contain a Carter subgroup but G satisfies (E). Let H be a minimal normal subgroup of G. Then H = T1 × . . . × Tk , where T1 ≃ . . . ≃ Tk ≃ T , and T is a finite simple group. By definition, G/H satisfies (E). Thus, by induction, there exists a Carter subgroup K of G = G/H. Let K be a complete preimage of K then K satisfies (E). If K 6= G then, by induction, K contains a Carter subgroup K ′ . Note that K ′ is a Carter subgroup of G. Indeed, assume that x ∈ NG (K ′ ) \ K ′ . Since K ′ H/H = K is a Carter subgroup of G, we obtain that x ∈ K. But K ′ is a Carter subgroup of K, thus x ∈ K ′ . Hence, G = K, i. e., G/H is nilpotent. If H is Abelian then G is solvable. Therefore G contains a Carter subgroup. So, assume that T is a non-Abelian finite simple group. First we will show that CG (H) is trivial. Assume that CG (H) = M is nontrivial. Since T is a non-Abelian simple group, it follows that M ∩ H = {e}, so M is nilpotent. By Lemma 2.1.2, we obtain that G/M satisfies (E). By induction, we obtain that G/M contains a Carter subgroup K. Let K ′ be a complete preimage of K in G. Then K ′ is solvable and, therefore, contains a Carter subgroup K. As above, we obtain that K is a Carter subgroup of G, a contradiction. Hence, CG (H) = {e}. Since H is a minimal normal subgroup of G, we obtain that AutG (T1 ) ≃ AutG (T2 ) ≃ . . . ≃ AutG (Tk ). Thus there exists a monomorphism ϕ : G → (AutG (T1 ) × . . . × AutG (Tk )) ⋋ Symk = G1 and we identify G, with Gϕ . Denote by Ki a Carter subgroup of AutG (Ti ) and by A the subgroup AutG (T1 ) × . . . × AutG (Tk ). Since G/H is nilpotent, the following representations are valid: Ki Ti = AutG (Ti ) and G1 = (K1 T1 × . . . × Kk Tk ) ⋋ Symk . Let πi : G ∩ A → (G ∩ A)/C(G∩A) (Ti ) be the canonical projections. Since G/(G ∩ A) is transitive, we obtain that (G ∩ A)πi = Ki Ti . Since AutG∩A (Ti ) = Ki Ti , then G ∩ A satisfies (E) and, by induction, contains a Carter subgroup M . By Lemma 2.2.3, we obtain that M πi is a Carter subgroup of Ki Ti . Therefore we can assume that M πi = Ki . In particular, if R = (K1 ∩ T1 ) × . . . × (Kk ∩ Tk ) then M ≤ NG (R). In view of Theorems 2.1.4 and 5.7.2, the Carter subgroups in each finite group are conjugate. Since (G ∩ A)/H is nilpotent, we get that G ∩ A = M H, so G = NG (M )H. Moreover, NG (M ) ∩ A = M , hence, NG (M ) is solvable. Since M normalizes R and M πi = Ki , we obtain that NG (M ) normalizes R, and so, NG (M )R is SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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solvable. Therefore it contains a Carter subgroup K. By Lemma 6.2.1, (K ∩ A)πi is a Carter subgroup of (NG (M )R ∩ A)πi (in this case, R plays the role of the subgroup T from Lemma 6.2.1), and so, (K ∩ A)πi = Ki . Assume that x ∈ NG (K) \ K. Since G/H = NG (M )H/H T = KH/H, it follows that x ∈ H. Therefore, xπi ∈ (NG (K) ∩ A)πi ≤ NTi ((K ∩ A)πi ) = Ki . Since i Ker(πi ) = {e}, it follows that x ∈ R ≤ NG (M )R. But K is a Carter subgroup of NG (M )R. Hence, x ∈ K. This contradiction completes the proof. 6.3. Example. In this Subsection, we will construct an example showing that we cannot substitute the condition (E) by a weaker condition: For each composition factor S of G, AutG (S) contains a Carter subgroup. This example also shows that an extension of a group containing a Carter subgroup, by a group containing a Carter subgroup, may not contain a Carter subgroup. Consider L = PSL2 (33 ) ⋋ h ϕ i, where ϕ is a field automorphism of PSL2 (33 ). Let X = (L1 × L2 ) ⋋ Sym2 , where L1 ≃ L2 ≃ L and if σ = (1, 2) ∈ Sym2 \ {e}, (x, y) ∈ L1 × L2 then σ(x, y)σ = (y, x) (the permutation wreath product of L and Sym2 ). Denote by H = PSL2 (33 ) × PSL2 (33 ) a minimal normal subgroup of X and by M = L1 × L2 . Let G = (H ⋋ h(ϕ, ϕ−1 ) i) ⋋ Sym2 be a subgroup of X. Then the following statements hold: 1. 2. 3. 4.

For every composition factor S of G, AutG (S) contains a Carter subgroup. G ∩ M E G contains a Carter subgroup. G/(G ∩ L) is nilpotent. G does not contain a Carter subgroup.

1. Clearly, we need to verify the statement only for non-Abelian composition factors. Every nonAbelian composition factor S of G is isomorphic to PSL2 (33 ) and AutG (S) = L. By Theorem 5.4.1, L contains a Carter subgroup (which is equal to a Sylow 3-subgroup). 2. Since (G ∩ M )/H is nilpotent, from the previous statement, we obtain that G ∩ M satisfies (E). So it contains a Carter subgroup (it is easy to see that a Sylow 3-subgroup of G ∩ M is a Carter subgroup of G ∩ M ). 3. Evident. 4. Assume that K is a Carter subgroup of G. Then KH/H is a Carter subgroup of G/H. But G/H is a non-Abelian group of order 6, hence, G/H ≃ Sym3 and KH/H is a Sylow 2-subgroup of G/H. By Lemma 2.1.2, AutK (PSL2 (33 )) is a Carter subgroup of AutKH (PSL2 (33 )) = PSL2 (33 ). But PSL2 (33 ) does not contain Carter subgroups in view of Theorem 5.4.1. 6.4. Classification of Carter subgroups . In view of condition (E) and Theorem 6.2.2, the description of Carter subgroups in finite groups is reduced to the classification of Carter subgroups in almost simple groups A, with A/F ∗ (A) nilpotent. The classification of Carter subgroups in groups with this condition is obtained in the previous Sections and we give it here for convenient usage. We first prove the following theorem showing that if, for a subgroup S of Aut(G), there exists a Carter subgroup then it exists in every larger group S ≤ A ≤ Aut(G) (here G is a known simple group). Theorem 6.4.1. Let G be a finite simple group and let A, with G ≤ A ≤ Aut(G), be an almost simple group with the simple socle G. Assume that A contains a subgroup S such that G ≤ S and S contains a Carter subgroup. Then A contains a Carter subgroup. Proof. Let K be a Carter subgroup of S. Clearly, we can assume that S = KG. Assume that either G is sporadic or G ≃ Altn for some n > 5. Since, by Lemma 3.2.11, each element of odd prime order of G is conjugate to its inverse, and since |Aut(G) : G| is a 2-power, Lemmas 2.4.2 and 2.4.6 imply that if some group G ≤ S ≤ Aut(G) contains a Carter subgroup K then K is a Sylow 2-subgroup of S. Since |A : S| is a 2-power, the statement of the theorem in this case follows from Lemma 2.4.6. Assume that G = 3 D4 (q). By [40, Theorem 1.2 (vi)], each element of G is conjugate to its inverse. If q is odd then Lemma 4.2.5 implies that K is a Sylow 2-subgroup of S. So, by Lemmas 2.4.6 and 4.2.5, it follows that A satisfies (ESyl2), i. e., contains a Carter subgroup. If q = 2t is even then, by Theorems 5.3.1 and 5.4.1, it follows that S = Aut(G) and we have nothing to prove. SIBERIAN ADVANCES IN MATHEMATICS

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Assume that G is a group of Lie type, G 6≃ 3 D4 (q) and if G ≃ D4 (q) then S ≤ A1 , where A1 ≤ Aut(D4 (q)) is defined in Theorem 5.3.1. Then one of statements (a)–(d) of Theorem 5.4.1 for S is valid. Consider all these cases separately. Assume that statement (a) for S is valid. In this case, we have |Aut(G) : S| 6 2, and so, for each A such that S ≤ A ≤ Aut(G), either A = S, or A = Aut(G). In any case, for A, statement (a) of Theorem 5.4.1 is valid and A contains a Carter subgroup. Assume that statement (b) for S is valid. Then |Aut(G) : S| = 2 and either A = S or A = Aut(G). b = PGL2 (3t ) satisfies (ESyl2). Hence, In the first case, we have nothing to prove. In the second case, G by Lemma 4.3.3, the group A also satisfies (ESyl2) and, by Lemma 2.4.3, contains a Carter subgroup. Assume that statement (c) of Theorem 5.4.1 for S is valid. Then S = Aut(G) and we have nothing b satisfies to prove. Assume that, for S, statement (d) of Theorem 5.4.1 is valid. By Lemma 4.3.1, S ∩ G b also satisfies (ESyl2). Hence, (ESyl2). By Lemma 4.3.3, every subgroup A of AutG containing S ∩ G by Lemma 2.4.3, it contains a Carter subgroup. Now, assume that G = D4 (q) and S satisfies the conditions of Theorem 5.3.1. Since graph automorphisms of orders 2 and 3 do not commute, only one of them can be contained in a nilpotent subgroup. Thus we can assume that only one of them is contained in A. Then, for every subgroup A containing S, there are valid the statements either of Theorem 5.3.1 or Theorem 5.4.1 (assertion (a) if q is even, and assertion (d) if q is odd), i. e., it contains a Carter subgroup. Note that from Theorem 6.4.1 and [28] the following interesting corollary follows. Lemma 6.4.2. Let S be a known finite simple group, S 6≃ J1 , and G = Aut(S). Then G possesses a Carter subgroup. Proof. By [28, Theorems 2 and 3], if S is not of Lie type and is not equal to J1 then the group of its automorphisms Aut(S) satisfies (ESyl2), and by Lemma 2.4.3, contains a Carter subgroup. Now, if S is of Lie type in even characteristic then Aut(S) contains a Carter subgroup in view of Theorem 5.4.1(a). If S is of Lie type in odd characteristic and S 6≃ 2 G2 (32n+1 ) then Sb satisfies (ESyl2), so, contains a Carter subgroup by Lemma 2.4.3. By Theorem 6.4.1, Aut(S) contains a Carter subgroup. Finally, if S ≃ 2 G2 (32n+1 ) then Aut(S) contains a Carter subgroup in view of Theorem 5.4.1(c). The tables below are arranged in the following way. In the first column, it is contained a simple group S such that the Carter subgroups of Aut(S) are classified. In the second column, we give conditions for a subgroup A of Aut(S) to contain a Carter subgroup. In the third column, we give the structure of a Carter subgroup K. In every subgroup of Aut(S) lying between S and A, the Carter subgroups do not exist. By Pr (G), a Sylow r-subgroup of G is denoted. We denote by ϕ a field automorphism of a group of Lie type S, and by τ a graph automorphism of a group of Lie type S contained in K (since graph automorphisms of order 2 and 3 of D4 (q) do not commute, only one of them can be in K). If A does not contain a graph automorphism then we suppose that τ = e. By ψ we denote a field automorphism of S of maximal order contained in A (it is a power of ϕ but hψi may different from hϕi). By K(U3 (2)), a Carter subgroup of order 2 · 3 of 2\ A2 (2) is denoted. If G is solvable then we denote by K(G) a Carter subgroup of G. In Table 8, by ζ we denote a graph-field automorphism of order 2t of A2 (22t ). Table 7. Groups of automorphisms of alternating groups containing Carter subgroups Group S

Conditions on A Structure of K

Alt5

A = Sym5

K = P2 (Sym5 )

Altn , n > 6

none

K = P2 (S)


CARTER SUBGROUPS OF FINITE GROUPS Table 8. Groups of automorphisms of classical groups containing Carter subgroups Group S

Conditions on A

Structure of K

A1 (q), q ≡ ±1 (mod 8)

none

K = NA (P2 (S))

A1 (q), q ≡ ±3 (mod 8)

Sb ≤ A

b K = NA (P2 (S))

K = hϕ, τ i ⋌ Sϕ2′

h S, ζg i ≤ A ≤ S ⋋ h ζ i,

K = hζgi × K(PGU3 (2))

An (2t ), t > 2 if n = 1 A2 (22t ), 3 ∤ t

ϕg ∈ A, g ∈ Sb

CA∩Sb(ϕ2′ ) ≃ PGU3 (2) An (q), q odd, n > 2

none

K = P2 (A) × K(O(NA (P2 (A))))

A2 (2t ), t odd, 3 ∤ t

hS, ϕ2′ gi ≤ A ≤ Sb ⋋ hϕ2′ i CA∩Sb(ϕ2′ ) ≃ PGU3 (2)

K = hϕ2′ i × K(PGU3 (2))

CA∩Sb(ϕ2′ ) ≃ PSU3 (2)

K = hϕ2′ i × P2 (PSU3 (2))

A2 (2t )

A = Aut(S)

K = hϕi ⋌ P2 (Sϕ2′ )

An (q), q odd

none

K = P2 (A) × K(O(NA (P2 (A))))

An (2t ), n > 3

A = Aut(S)

K = hϕi ⋌ P2 (Sϕ2′ )

B2 (q), q ≡ ±1 (mod 8)

none

K = P2 (A) × K(O(NA (P2 (A))))

B2 (2 ), t > 2

ϕ∈A

K = hϕ, τ i ⋌ P2 ((Sτ )ϕ )

B2 (q), q ≡ ±3 (mod 8)

K = P2 (A) × K(O(NA (P2 (A))))

Bn (q), q odd, n > 3

Sb ≤ A none

K = P2 (A) × K(O(NA (P2 (A))))

Cn (q), q ≡ ±1 (mod 8)

none

K = P2 (A) × K(O(NA (P2 (A))))

Cn (q), q ≡ ±3 (mod 8)

K = P2 (A) × K(O(NA (P2 (A))))

Cn (2t ), n > 3

Sb ≤ A

A = Aut(S)

K = hϕi × P2 (Sϕ2′ )

D4 (q), q odd

none

if |τ | 6 2 then

2

2 2 2

t

K = P2 (A) × K(O(NA (P2 (A)))); if |τ | = 3 then K = hτ, ψi ⋌ P2 (Sτ ) D4 (2t )

if |τ | 6 2 then

ϕ∈A

K = hτ, ϕi ⋌ P2 (Sϕ2′ ); if |τ | = 3 then K = hτ, ϕi ⋌ P2 ((Sτ )ϕ2′ ) Dn (q), q odd, n > 5

none

K = P2 (A) × K(O(NA (P2 (A))))

Dn (2t ), n > 5

ϕ∈A

K = hτ, ϕi ⋌ P2 (Sϕ2′ )

Dn (q), q odd

none

K = P2 (A) × K(O(NA (P2 (A))))

Dn (2t )

A = Aut(S)

K = hϕi ⋌ P2 (Sϕ2′ )

2

2

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VDOVIN Table 9. Groups of automorphisms of sporadic groups containing Carter subgroups Group S

Conditions on A Structure of K

J2 , J3 , Suz, HN

A = Aut(S)

K = P2 (A)

6≃ J1 , J2 , J3 , Suz, HN

none

K = P2 (A)

Table 10. Groups of automorphisms of exceptional groups of Lie type containing Carter subgroups

2

Group S

Conditions on A

Structure of K

B2 (22n+1 ), n > 1

A = Aut(S)

K = hϕi × P2 (2 B2 (2))

(2 F4 (2))′

none

K = P2 (A)

F4 (22n+1 ), n > 1

A = Aut(S)

K = hϕi × P2 (2 F4 (2))

A = Aut(G)

hϕi ⋌ (2 × P ),

2

2

G3 (32n+1 )

where |P | = 3|ϕ|3 remaining, q odd

none

K = P2 (A) × K(O(NA (P2 (A))))

remaining, q = 2t

ϕg ∈ A, g ∈ Sb

hτ, ϕi ⋌ P2 (Sϕ2′ )

ACKNOWLEDGMENTS I am grateful to my scientific advisor, Corresponding member of the Russian Academy of Sciences V. D. Mazurov. His contribution to my development as a mathematician and his permanent support are inestimable. I am also sincerely thankful to Professor M. C. Tamburini who initiated my work on these problems, for her support. I especially thank Professor A. V. Vasiliev, Doctor M .A. Grechkoseeva, Doctor A. V. Zavarnitsine, and Doctor D. O. Revin for very useful discussions on the topic of the paper which allow to simplify some proofs and to eliminate inaccuracies and mistakes. I am also pleased to thank Professor A. S. Kondratiev for valuable comments which improved the final version of the paper. I am grateful to and wish to honor the blessed memory of Professor Yu. I. Merzlyakov who awaken my interest to Algebra and Group theory. A part of the research was carried out during my post-doctoral fellowship at the University of Padua (Italy) and I am grateful to all members of Algebra chair, especially, to Professor F. Menegazzo for the support. The research was partially supported by the Russian Foundation for Basic Research (grant 99–01– 00550, 01–01–06184, 02–01–00495, 02–01–06226, and 05–01–00797), by the State Maintenance Program for Young Russian Scientists and the Leading Scientific Schools of Russian Federation (grant MK–1455.2005.1 and MK–3036.2007.1), by Siberian Division of the Russian Academy of Sciences (grant N 29 for young scientists and Integration Project 2006.1.2), and by the Program “Universities of Russia” (grant UR.04.01.202). REFERENCES 1. A. Borel et al., Seminar on Algebraic Groups and Related Finite Groups, Lecture notes in Mathematics 131 (1970). ´ de rang maximum des groupes de Lie clos,” 2. A. Borel and J. de Siebental, “Les-sous-groupes fermes Comment. Math. Helv. 23, 200–221 (1949). 3. R. W. Carter, “Nilpotent self-normalizing subgroups of soluble groups,” Math. Z. 75, 136–139 (1961). 4. R. W. Carter, “Conjugacy classes in the Weyl group,” Compos. Math. 25 (1), 1–59 (1972). 5. R. W. Carter, Simple Groups of Lie Type (John Wiley and Sons, 1972). SIBERIAN ADVANCES IN MATHEMATICS Vol. 19 No. 1 2009


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