May 11, 1997 - pendently and jointly investigatedby Bachmuth and Mochizuki in a series of ... Manufactured in the United States of America ..... H fq {g V < } f3.
ILLINOIS JOURNAL OF MATHEMATICS Volume 42, Number 2, Summer 1998
AUTOMORPHISMS OF METABELIAN GROUPS WITH TRIVIAL CENTER R(JDIGER GOBEL AND AGNES T. PARAS
1. Introduction
Let F(n) denote the free group of rank n and let B(n) F(n)/F(n)", the free metabelian group of rank n. The automorphism group Aut(B(n)) has been independently and jointly investigated by Bachmuth and Mochizuki in a series of papers dating from 1965 to 1987. In ], the outer automorphism group Out(B (2)) is shown to be isomorphic to G L2 (Z). When n 3, Aut(B (n)) has been shown to be infinitely generated in [2]. For n > 4, they showed [3] that Aut(F(n)) --+ Aut(B(n))
-
1;
i.e., every automorphism of B(n) is induced by an automorphism of F (n) and hence, Aut(B(n)) is finitely generated. This is carried out using the faithful Magnus representation of IA(B(n)) as a subgroup of GLn(Z[F(n)/F(n)’]) (IA(G) is the normal subgroup of Aut(G) consisting of automorphisms of G which induce the identity on the quotient G! G’), and ideas and methods influenced by matrices and matrix groups over integral Laurent polynomial rings. Instead of considering the automorphism group of a given metabelian group, we propose to approach the problem from the opposite direction, namely: Which groups can be realized as the automorphism groups of metabelian
groups? That is, for which groups H does there exist a metabelian group G such that Aut G is isomorphic to H? The case when G is a torsion free, nilpotent group of class 2, hence metabelian with non-trivial center, has been considered by Dugas and G0bel. In [8], they adapt Zalesskii’s matrix construction of a torsion free, nilpotent group of rank 3 and class 2 with no outer automorphisms, to show that any group H can be realized as
Aut(G)/Stab(G)
H
Received May 11, 1997. 1991 Mathematics Subject Classification. Primary 08A35,20C07, 20F29; Secondary 04A20, 20K20. This work was done at the University of Essen, where the first author was supported by the GermanIsraeli Foundation for Scientific Research and Development and the second author was supported by the German Academic Exchange Servie (DAAD). (C) 1998 by the Board of Trustees of the University of Illinois Manufactured in the United States of America
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RODIGER G(BEL AND AGNES T. PARAS
334
for some torsion flee, nilpotent G of class 2, where
Stab(G)
{c
6
Aut(G)
ot
induces the identity on Z(G) and G/Z(G)}.
Notice that in this setting, Inn(G) _c Stab(G) and Stab(G) is abelian. Using the Baer-Lazard theorem, which provides a correspondence between nilpotent groups of class 2 and alternating bilinear maps, they arrive at a similar result [9] with the added information that Stab(G)/Inn(G) is isomorphic to a direct sum of IGI-copies of the cyclic group Z2 of order 2. In this paper, we consider metabelian groups with trivial center. If a group G has trivial center, then G automatically embeds as Inn G in Aut G. The quotient Aut(G) ! Inn(G) is usually called Out(G), the outer automorphism group of G. Hence we ask:
For which groups H does there exist a metabelian group G with trivial H? center such that Out G
A partial answer is supplied below by our main results. A group is said to be complete if its center and outer automorphism group are both trivial. In [11], Gagen and Robinson classified all finite, metabelian and complete groups. Using homological methods, Robinson considered infinite soluble and complete groups in [20]. Here, we consider, the infinite metabelian case and provide infinitely many non-isomorphic complete, metabelian groups in the following theorem.
.
THEOREM. Let B be a free metabelian group of rank ) with 3 < ,k < 2 Then there exists a torsion free, complete, metabelian group G embedding B, with G containing an abelian and characteristic subgroup A of cardinality 2 such that
G/A
B/B’.
It is interesting to note that this non-abelian result is obtained by mainly applying methods from abelian group theory, properties of group rings and the Magnus representation of a free metabelian group. A group is said to be a unique product group (UP group) if, given any two nonempty finite subsets A and B of G, there exists at least one element x of G that has a unique representation in the form x ab with a 6 A and b 6 B. Free groups and, more generally, right ordered groups are examples of UP groups. It is a well-known fact that no group can have its automorphism group be cyclic of odd order > or infinite cyclic (see Robinson 19], [21 and Pettet 16], 17], 18] for other examples of non-automorphism groups). In contrast, we see from our second main result that every abelian group is isomorphic to the automorphism group of some metabelian group modulo its inner automorphisms. THEOREM. Every abelian group and every UP group can be realized as the outer automorphism group of some metabelian group with trivial center
METABELIAN GROUPS WITH TRIVIAL CENTER
335
The construction is founded on the endomorphism rings of torsion free abelian groups, the extraction of the multiplicative units of a ring and the semi-direct product of two abelian groups.
2. Representation of free metabelian groups
For convenience, we include two well-known results concerning free soluble groups and matrix representation of groups, which are due, respectively, to Sm61kin and Magnus. The following is a special case of a lemma due to Magnus 14], which is referred to in the literature as Magnus representation and has proved to be a useful tool in various contexts. I }. Let Suppose F is a non-cyclic free group with basis {xi
{S
X
F’
I} and {ai
xiF"
I}
be generators of F/F’ and F/F" respectively. Let islZ[F/F’]ti be a free Z[F/F’]module of rank II l. The set of matrices
F/F’
0
iel[F/F’]ti
F/
g
Zil riti
riti
il
il
forms a group under formal matrix multiplication.
LEMMA 2.
14]. The map
aJ--+[ sjtj Oil extends to an injective homomorphism
"
F/F"--
F/F’
0
isiZ[F/F’]ti
If B is a metabelian group, then/ B!B’ acts on B’ via conjugation. Hence there exists a homomorphism q)" B Aut(B’). This extends to a ring homomorphism q)"
Z[/]
-
End(B’),
and so B’ can be viewed as a Z[B]-module. In the case when B is free metabelian, the Magnus representation enables us to see that each nonzero element of q)(Z[/}]) C End(B’) is a monomorphism. We express this in terms of modules"
RODIGER G6BEL AND AGNES T. PARAS
336
F/F". Then B’ is a torsion free Z[]-module.
COROLLARY 2.2. Let B
Proof Using the Magnus representation, we identify B with p(B) and notice that B’ embeds in
[
C
o
t
iIZ[F/F’]ti
o) b=--nj( O) ( Ynjuj 0)
If a
y
over, if
6
B and Z
6
z
B’, then Z a
uj vj
then
zb
z
,wherenj
B
-
is the canonical epimorphism. Let z Z[F/F’] is an integral domain (see [12]),
0] zx
o)
More-
Z[B],
Z, uj
F/F’ and
B/B’
ZiI biti,
where bi
(Z biti) (Z njblj) =0iff b =0or njuj Hence B’ is a torsion free Z[/)]-module.
t
Z[F/F’]. Since 0 for each/.
[21
From now on, we identify B’ with a subgroup of )ielZ[]ti Suppose S is a ring and G is a group. Let I (S, G) denote the augmentation ideal of S[G], which consists of all x 0. Let il biti correspond to an ngg S[G] such that Y ng element of B’. A characterization of the bi’s is given in ], but for our purposes it is enough to know that each bi is an element of I (Z,/). Since the action of b 6 Z[/] on z bbi ti and the commutator equality bi ti B’ is defined by z b
Y
[uv, w]
[u, willy, w]
holds for elements of any group, it suffices to verify that the free generators 7z(ai) satisfy
[Tz(ai), (aj)] (sj 1)ti + (1 si)tj. Hence B’ is a subgroup of i11 (Z, B)ti. Let F d) denote the d-th derived subgroup of the non-cyclic free group F.
THEOREM 2.3 (Sm61kin [22]). Let G F/F d) be a free soluble group and ct Aut(G) which is the identity when restricted to F(d-l)/F (d). Then t Inn G. In fact, c is conjugation by an element of F(d-1)/F (d) The following corollary follows easily.
337
METABELIAN GROUPS WITH TRIVIAL CENTER
COROLLARY 2.4. Suppose B F/ F", where F is a free group and B embeds in Aut(B) such that otln, 1’ Then
a group G. Let or,
(i) (ii)
ot
ot
/3. Inn B and extends to G if and only if extends to G. 3. Extensions of free metabelian groups
0, for some prime Recall that an abelian group A is p-reduced if ’nEo9 P nA number p. If an abelian group A is p-reduced and torsion free, we denote its completion relative to the p-adic topology by A. Note that conju,g.ation by b 6 B extends Suppose B is free metabelian, B’ < H < B’ and H is B-invariant, uniquely to H. B of elements of i.e., closed under conjugation by elements of B. The set G the form h b forms a group under the operation
".
(h. b)(g. c)
hg b-
bc where h, g
6
H, b, c
6
B.
Note however that representation of elements of G in the form h b is not unique; i.e., G is not a semi-direct product.
LEMMA 3.1. Suppose G H.B, where B is free metabelian, B’ < H < B’ and H is B-invariant. If A is a normal, abelian subgroup of G then A < H. Hence H is the largest normal, abelian subgroup of G and so is characteristic in G.
Proof We first observe using Corollary 2.2, that if b 6 B and x 6 B’ with 1; i.e., conjugation by elements b 6 B \ B’ does x b x, then either b 6 B’ or x not leave non-trivial elements of B’ fixed. By the continuity of homomorphisms on H and the B-invariance of’H, it follows that conjugation by b 6 B \ B’ does not leave non-trivial elements of H fixed. h.b, h 6 H and b 6 B \ B’. Suppose there exists x 6 A such thatx e" A for all c B; Since A is normal, abelian in G, x A and xe’x x-’ i.e., [c,x -] [x,c]. Hence [c,x] [c,x]. Since[c,x] 6 Handh 6 H, [c, x]. Taking c b, we get [b, h] b [b, h]. This implies [b, h] 1, [c, x] bb 1. Sox b (B \ B’) O A and h. Since b 6 B \ B’, it follows that h i.e., h A for all c B. Since be’b -1 A f3 B’, b. be’b -l be’b b b e’. This means b for all c 6 B. But this occurs (bC) b-’ b e’, and so, by our first observation, b only if b 1, thus giving us a contradiction.
-
x-x
-
Throughout this section and the next, we adopt the following notation"
R
R*
z[?] the group of multiplicative units of R
RODIGER GOBEL AND AGNES T. PARAS
338
the image of the element x under the homomorphism q9 the R-submodule generated by x the p-height of x in G, i.e., the largest integer k such that
hp
pk divides x
6
G in the torsion free abelian group G
LEMMA 3.2. Suppose B is free metabelian with rank at least three and p Aut(B’). If e e R for every e in every basis of B’, then 99 R.
Proof If B has rank at least three, B’ has at least two R-independent elements, say f. Suppose e e and f if, where r, s 6 R. Then (e. f) (e. f)t for some 6 R, since e. f is in some basis of B’. Because q9 6 Aut(B’), (e. f)t e "fs. It follows from the R-independence of e and f and Corollary 2.2 that s. r e and
Now consider an arbitrary element x of some basis of B’. Then either {x, e} or {x, f} is R-independent. By the foregoing argument, it is clear that x x r. Moreover R*. 99
B’ and 99 Aut(B’) such that e e R and h pB’ (e) # 0 for 0 for all and hpB’ (eo) some prime p, then there exists eo B’ such that that B’; Moreover such 0 and is B’ in x n x i.e., pure primes p. # if LEMMA 3.3. If e
e.
then x
e e
e
Proof
Write e
p. Take e0
f.
fn (f
B’, 0 -fin
e
Z) such that h pB (f)
0 for all primes
If A is a subgroup of an abelian, torsion free group G, we define the pure subgroup generated by A to be
(A),={x 6G’x 6A for someO#n6Z}.
_
We use Jp to denote the group (or ring) of p-adic integers. Lemma 3.4 is an adaptation of Lemma 3 in [5], which is a result in a strictly abelian setting, and will be used often in the constructions of the next propostion and section.
.
.
LEMMA 3.4. Suppose B is free metabelian of rank with 3 < ,k < 2 Let H be a pure, B-invariant subgroup ofB with cardinality less than 2 and B’ < H < If is a monomorphism F is a subset of B with cardinality less than 2 n’’ and o B’ such that H f3 F 0 and PlB’ R*, then there exists g B such that gO (H, gR).
’
and F
(3
(H, gR).
Proof. H. IfH
O.
"’
’.
Since 0 is defined on B’, p extends to B’ by continuity and restricts to H, then there exists g 6 H such thatg q IH, gR). H and
_
339
METABELIAN GROUPS WITH TRIVIAL CENTER
(H, gR). f3 F
0. Suppose H
H. By Lemma 3.2, OIB,
.
R* implies there
exists a basis element e of B’ such that e e R. Using Lemma 3.3, we can assume that h pBt (e) 0 for every prime p and, hence, e R is pure in B’. Since HI < 2 there exists 6 Jp such that H f3 H (see [7]). Let g e If g (H, gR)., then there exists a non-zero n 6 N such that gn (g)n h gr for some h 6 H, r 6 R. So (en-r) G H t3 H and e n e r. Since e g is pure, e 6 e R, which is a contradiction. Therefore g (H, gg).. There exist 2 algebraically independent ’s in Jp such that H f3 H 1. So there exist 2 g’s (g e ) such that g q; (H, gR), =. Hg. Suppose each such Hg has Hg N F :/: 13; i.e., there exists x 6 F such that x h gr for some non-zero n 6 1, h 6 H and r 6 R \ {0}. Since there are less than 2 choices for quadruples (n, x, h, r) and 2 s choices for g, there must exist distinct and # with He and He,, such that x 0. h e r and x h e lzr. The two equations yield #, since r This contradiction leads us to conclude that there exists g 6 B’ such that Hg f3 F 0 and g Hg. D
’
’
-
Remark 3.5. The proof of Lemma 3.4 goes through if we restrict our choice of ’s to any subset of E of Jp containing 2 algebraically independent elements.
PROPOSITION 3.6. less than 2
If B is free metabelian with rank at least three, ) a cardinal
and
{1 # o, Out(B) o < .}, then there exists a torsion free, metabelian group G into which B embeds such that each p does not extend to an automorphism of G.
Proof Suppose B is of rank less than 2 s. Apply Lemma 3.4 to H0 B’, F0 0 6 B’ such that 80 (B’, gff).. Suppose Ha (B’, gff fl < % , ). C B F {g fl < } and ln’ such that H & F D. By Lemma 3.4, there exists g 6 B’ such that g D. (H, gff). =" H+ and H+l F
’
and tP01, to obtain go
4, Trans. Amer. Math. Soc. 292 (1985), 81-101. A.L.S. Corner and R. Gtibel, Prescribing endomorphism algebras--A unified treatment, Proc. London Math. Soc. (3) 50 (1985), 447-479. Essentially rigid floppy subgroups of the Baer-Specker group, to appear in Manuscripta
6.
Mathematica. M. Dugas and R. Gtibel, Every
104.
3. 4.
7. 8. 9. 10. 11.
12. 13. 14.
cotorsion-free algebra is an endomorphism algebra, Math. Z. 181 (1982), 451-470. Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. (3) 45 (1982), 319-336. Torsion free nilpotent groups and E-modules, Arch. Math. 54 (1990), 340-351. Automorphisms of torsion free nilpotent groups of class 2, Trans. Amer. Math. Soc. 332 (1992), 633-646. R.H. Fox, Free differential calculus I, Ann. of Math. (2) 57 (1953), 547-560. R.M. Gagen and D. J. S. Robinson, Finite metabelian groups with no outer automorphisms, Arch. Math. 32 (1979), 417-423. G. Karpilovsky, Commutative group algebras, Marcel Dekker, New York, 1983. Unit groups of classical rings, Clarendon Press, Oxford, 1988. W. Magnus, On a theorem of Marshall Hall, Ann. of Math. (2) 411 (1939), 764-768.
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RODIGER GBEL AND AGNES T. PARAS
15. H.Y. Mochizuki, Automorphisms of solvable groups, Part H, C. M. Campbell and E. F. Robertson, ed., London Math. Soc. 121, Cambridge Univ. Press, Cambridge, (1987), 15-29. 16. M.R. Pettet, Locally finite groups as automorphism groups, Arch. Math. 48 (1987), 1-9. 17. Almost-nilpotent periodic groups as automorphism groups, Quart. J. Math. Oxford (2) 41 (1990), 93-108. 18. Free-by-finite cyclic automorphism groups, Rocky Mountain J. Math. 23 (1993), 309317. 19. D.J.S. Robinson, Infinite torsion groups as automorphism groups, Quart. J. Math. (2) 30 (1979), 351-364. 20. Infinite soluble groups with no outer automorphisms, Rend. Sem. Mat. Univ. Padova ti2 (1980), 281-294. 21. Groups with prescribed automorphism group, Proc. Edinburgh Math. Soc. 25 (1982), 217-227. 22. A. L. Sm61kin, Two notes on free soluble groups, Algebra Logika ti (1967), 95-109.
Rtidiger G6bel, Fachbereich 6, Mathematik und Informatik, Universitit Essen, 45117
Essen, Germany R.Goebel @ Uni-Essen.De
Agnes T. Paras, Department of Mathematics, University of the Philippines at Diliman, 1101 Quezon City, Philippines agnes @ math01 .cs.upd.edu.ph
