Primitive elements of the free groups of the varieties

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KEY WORDS: variety of groups, free group of a variety, primitive element, normal closure, ... r > rn, we can choose r - rn elements from the set {xl,..., x,-} that freely ...
Mathematical Notes, 11"oi.61, No. 6, 1997

P r i m i t i v e E l e m e n t s o f t h e Free Groups of t h e V a r i e t i e s 92fftn E . I. T i m o s h e n k o

U D C 512.54

ABSTRACT. For groups of the form FIN', we find necessary and sufficient conditions for an element g E N/N' to belong to the normal closure of an element h E FIN ~. It is proved that, in contrast to the case of a free metabelian group, for a free group of the variety 9.1012,there exists an element h whose normal closure contains a primitive element g, but the elements h and g:i:l are not conjugate. In the group F(P.lff[2), two nonconjugate elements are chosen that have equal normal closures. KEY WORDS:

variety of groups, free group of a variety, primitive element, normal closure, theorem on freedom.

Let !DI b e a variety of groups and let Fr(gX) be a free group of the variety of rank r . Sometimes we shall omit the subscript and denote this group by F(~JI). Denote by (x, V, .--, z> the subgroup generated by x, y, ... , z and by (x, y, ... , z)G the normal subgroup of the group G generated by these elements. A n element g of F(~JI) is said to be primitive if it can be included in a system of generators of the group F(~JI). A system of elements gl,-.., gm ( m F(0x) contains a primitive element h? If 9/I is the variety of all groups, then it is well known that the element g is conjugate to one of the elements h • . A similar element g is conjugate to one of the elements h +1 . For free m e t a b e l i a n groups, the same result was recently o b t a i n e d b y Evans [1]. T h e following question arises: Does a similar t h e o r e m hold for free groups of varieties close to the variety 9.12 of metabelian groups? We consider free groups of finite rank of the variety 9.19In, where fftn is t h e variety of nilpotent groups of degree _< n. Let us n o t e that F(P.19~,,) = F / [ V n + I ( F ) , 7 , , + l ( F ) ] , where F is the free group with basis x l , . . . , x r , 7 1 ( F ) = F , a n d 7 n + l ( F ) = [7,,(F), F ] . As is known, these groups have some properties close to those of the m e t a b e l i a n groups. For instance, they are residually finite a n d satisfy the ascending chain condition for n o r m a l subgroups. At the same time, t h e y have some distinctions, which manifest themselves in problems treated in the present p a p e r a l r e a d y for n = 2. For instance, C. K. Gupta, N. D. G u p t a , a n d G. A. Noskov [2] proved that, for each e p i m o r p h i s m ~ : Fr(922) --* Fro(P22), r > m , there exist a u t o m o r p h i s m s a E Aut(Fr(P22)) and /~ E Aut(Fm(P-12)) such that a~/~ is the standard epimorphism 7r: F~(P.I 2) ~ F,,,(p.12). However, as was shown b y Evans, for t h e free groups of the varieties 9-1912, a similar s t a t e m e n t is no longer true. For each r , r _> 2, Evans i n d i c a t e d an epimorphism r F~+1(PAff[=) ---* F~(929~2) such that k e r C r is not the normal closure of a single element. We say that in a variety of groups if)I, the generalized theorem on freedom holds if for any group G = ( x l , . . . , x~ ; f l = 1, . . . , f m = 1 ; if)I} finitely presented in this variety and satisfying the condition r > rn, we can choose r - rn elements from the set { x l , . . . , x,-} t h a t freely generate a subgroup of the form F~_,~(gX) in G . As is known, the generalized t h e o r e m on f r e e d o m holds in the varieties of all groups [3], of solvable groups of degree < I [3], of nilpotent groups 9~,, [4], of centrally metabelian groups [5], and also for t h e varieties 9.1912, 91292 [6], and 9.1912 A ffl29.1 [7]. The following p r o p o s i t i o n shows that if the generalized t h e o r e m on f r e e d o m holds in a variety, then in the free groups of this variety, the primitive systems have maximality properties (in a certain sense). Translated from Malemalicheskie Zameiki, Vol. 61, No. 6, pp. 884-889, June, 1997. Original article submitted December 19, 1995. 0001-4346/97/6156-0739518.00

(~1997 Plenum Publishing Corporation

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P r o p o s i t i o n 1. A s s u m e that in a variety 93~, the generalized theorem on freedom holds. Let h i , ... , h,, be a primitive system of the group Er(gX), r >_ m . If h i , . . . ,hm 6 ( g l , . . . ,gin) F ' ( ~ ) , then ( h i , . . . , hm) F'(9~) = (gl, . . . , gin) F'(~). P r o o f . W i t h o u t loss of generality, we may assume that h / i s the image of an element xi of F . Assume that the n o r m a l closure of the elements gl, . . . , gm strictly contains (hi, . . . , hm) f(mz) 9 T h e n there exists w ( x m + l , . . . , xr) 6 F whose image is a nonidentity element of ( g l , . . . , gin) y(mz). On the other hand, in the group Fr(9)~) we can choose r - m elements from the basis induced by the basis x l , . . . , xr of the group F , such t h a t the subgroup generated by these elements is disjoint with ( g l , . . . , grn) F(9~). The contradiction t h u s obtained completes the proof of the proposition. [] Applying the result of Shmel'kin on the conjugacy of elements of a certain form that generate the same normal subgroup, we can readily obtain the above result of Evans from Proposition 1. T h e o r e m (M. Evans). Let g and h be elements of a free metabelian group S~ and let g be primitive. If g 6 (h) s" , then the element g is conjugate to one of the elements h q-1 . P r o o f . Let Sl, . . . , s~ be a basis of the group S~. Without loss of generality, we m a y assume that g = sx and h = Sl c, c 6 S~. Since the generalized theorem on freedom holds in the variety 9.12, it follows that (g)S, = (h)S,. T h e n it follows from Theorem 2 in [8] that the elements g and h are conjugate. This proves the theorem. [] Denote by Oig the Fox ith left derivative of an element g 6 F . Consider the natural h o m o m o r p h i s m : Z F --* Z F , where F = F I N . T h e image of the derivative Oig in the ring Z F is called the value of this derivative in the ring Z F . For an element of a free metabelian group, the following primitivity criterion holds [2, 9-11]. The image of an element g 6 F in the group F / F " is primitive if and only if the vector ( O ~ g , . . . , O~g) of the values of the derivatives in the ring Z ( F / F ' ) is unimodular. T h e following assertion is a consequence of the Evans theorem. However, its proof can be applied in other cases in which the primitivity criterion holds. For instance, C. K. G u p t a and the author proved that the primitivity criterion holds for the free groups of the varieties 92mPl,, where 9.1,, is the variety of Abelian groups of exponent dividing m . The case m -- 0 and n = 0 will be treated in a forthcoming paper. Therefore, Proposition I holds for the free groups of these varieties. Proposition

2. Suppose that g and h are elements of F~ and their images g' and h' in the group

Er(9~nPl) satisfy the relation

h'

6

(g,)F(91.~t).

If the element h' is primitive, then g' is also primitive. P r o o f . As is known, if the elements g l , . . . , g r generate the group G m o d u l o a nilpotent normal subgroup, t h e n these elements generate the group G. Therefore, it suffices to prove that the element gF" is primitive. It follows from the relation h e

(g)F(modE")

that the derivatives Oih belong to the idea2 generated by the elements 01g, . . . , Org together with the element y - 1. T h e bar over a symbol means that the corresponding element is from Z ( F / F ' ) . Ho~vever, we have r

E Oig('zi-1) =

~ " - 1,

i=1

and therefore, the ideal generated by the elements Olg, . . . , Org, contains the coordinates Oih of a unim o d u l a r vector. Hence, the element g F " is primitive. This completes the proof of the proposition. []

740

R e m a r k . If the n o r m a l closure of the elements g l , - - - , gm of a free m e t a b e l i a n group S~, r _> m , contains a primitive s y s t e m h i , . . . , hm, then, in general, gl, . - . , gm need not be a primitive system for m > 1 For example, consider a free metabelian group $2 with basis sl s2 Let a l = S l S ~ , a 2 = 2S~I be the corresponding basis of the group $2/S~. Choose c 6 S~ so that the element (a2 - 1 ) 0 2 c + 1 of the ring Z(S2/S~) is noninvertibte. T h e elements gl -- s~ and g2 -- s~ satisfy the relation s~, s2 E (gl, g2>s2 . However, the s y s t e m g l , g2 is not primitive, because the matrix formed by the derivatives Oigj, i, j = 1,2, is not invertible. 9

,

9

Consider the m a p p i n g fl of a free group F into the group of matrices M ( F / N )

(~

1)'

S

of the form

(1)

where ~ is the image of a n element g of F in the group F = F / N and ~ is an element of the free ring Z F of the m o d u l e T w i t h basis t l , . . . , t r . On the generators x l , . . . , x r , the mapping ~ is defined as follows:

where i ---- 1 , . . . , r . This m a p p i n g is the well-known Magnus embedding of the group F / [ N , N] into the m a t r i x group M ( F ) , i.e., kerfl -- N ' . Let g be an element of F . Its image is a m a t r i x of the form (1), and a --- a l t l + --- + ~ , t , , ~i E Z F . Denote by /g the ideal generated by the element ~ - 1 in the ring Z F . As is known, a i ----0ig. L e m m a . Assume that h 6 N and G = (g}F. Then hN' 6 G N ' if and only if there exist elements E Z F and ~a, ... , ~ 6 Ig such that Oih = 7ai + -7i. m

P r o o f . T h e n a t u r a l group h o m o m o r p h i s m Y --~ -ff/{gN> F can be extended to a h o m o m o r p h i s m r of the group M ( F ) onto the group M ( . ~ ) , where _~ = -F/(gg) y . T h e groups M ( F ) and M ( F ) are isomorphic to the w r e a t h products of a free Abelian group A of r a n k r with the groups T and _~, respectively. As is k n o w n (see, e.g., [12]), k e r r coincides with the group [B1, A]B1, where BI is the subgroup of the group M ( F ) t h a t consists of matrices of the form

and A is the group of all matrices of the form

where ~ E T . Thus, R = k e r v belongs to the normal closure of the m a t r i x

0) in the group M ( F ) . O n t h e other hand, the image of the matrix Mg u n d e r the action of the h o m o m o r p h i s m 7 is the identity element. Therefore, R coincides with the normal closure of the m a t r i x Mg in the group M(F). Let us verify that, for an element f of F , the conditions f N ' 6 G N ' and ( f N ' ) ~ E (aN') are equivalent. Indeed, a s s u m e that f~r 6 G #" . T h e n f a is an element of G # R . Hence, there exist z E G # and r 6 R such t h a t f # = zr. However, in this case r = z - i f ~ are elements of R N F # . All the more 741

the element r belongs to the intersection R1 tq F # , where R1 is the normal subgroup of the group M ( F ) that is generated by the matrices Mg and 1

Romanovskii [13] proved t h a t R1 n F # = G # , and therefore, r E G # . Hence, f # E G # . Now we consider an element h E N and find necessary and sufficient conditions that h #~ E G #r . Since the m a t r i x Mg is taken to the identity matrix under the h o m o m o r p h i s m 7, it follows t h a t G #r is generated, as a n o r m a l subgroup of F ar , by a single matrix of the form ( 10. 1) .

T h e normal closure of a

matrix of this form in the groups F #~ and M(ff) is the same. Let M~ = ( 0 be the image of the m a t r i x Ma in the group M ( f f ) , where ~ = ~:tl + - ' - + 3 r t r , and ~i is the image of an element a~ in the ring Z F . T h e normal closure of the matrix M~ in the group M ( F ) consists of all matrices of the form

(:o

where ~ is an arbitrary element of Z F . The image of the element h in the group M ( f f ) is t h e matrix A

1 0

A

A

/

Ozht~ + 0 2 h t 2 + ' " + O r h t r 1

"

Therefore, h a~ E G #~ if a n d only if there exists a ~ E Z_~ such that A

Oih = ~ai

(2)

for each i, i = 1, ... , r. T h e kernel of the natural h o m o m o r p h i s m system (2) is solvable in Z_~ with respect to "~ if and only if we have Oih = 7 ai

This completes the proof of the lemma.

: Z F ~ Z-~ is Ig, and hence the

"[- 7i"

[]

Now we can show that the Evans theorem fails for the variety P29~2. T h e o r e m . There is an element g o f Fr whose image in the group Fr(O.lcJ~2) generates a normal subgroup containing the image o f the element x: , but the d e m e n t s g and xl , as we11 as the elements g and 2771 , are n o t conjugate m o d u l o [Ts(F), 7s(F)]. P r o o f . We take for g the element g = X 1 [271, 272, X2, 271, 272]. Note that c = [27:, x2, x2, x : , 272] is an element of v a ( F ) . Therefore, to verify the relation 271 E (g)F modulo [va(r), 73(F)], it suffices to establish that a l c = 7(1 q- ~laZC) + 7 1 ,

0 2 c = ff:102c-b 72,

for some "7, 7 : , - - - , :r from Z ( F / ' 7 3 ( F ) ) , where : : , . . . , aic=

(~-I

_

l)(~-i

i.e., aic E I x , . Therefore, in (3) we can set 5 = 0. 742

... ,

arc = 7 : : a r c + 7 r

:r E Ix,. However, we have

_ 1)ai[x], x2, x2],

(3)

Now let us verify t h a t t h e e l e m e n t c c a n n o t be w r i t t e n in the f o r m [xl, u] in the g r o u p Fr(919~2). It suffices to c o n s i d e r t h e g r o u p of r a n k r = 2 a n d take the element u f r o m 73(F2). Let us c a l c u l a t e t h e values of t h e derivative 01 on the elements Ix1, u] a n d c in t h e ring Z ( F / 7 3 ( F ) ) . We h a v e 01[Xl, u] = (1 -- 5~1)01u,

alC = y-1(~21 -- 1 ) ( ~ -1 - 1 ) ( ~ -1 - 1)~71(~21 -- 1),

where y = Ix1, x2].

Each element of the ring Z(F/V3(F)) can be uniquely written in the form ZgJ

,

(4)

where j takes finitely m a n y i n t e g r a l values a n d gj -= gj(y, 51) is a p o l y n o m i a l over t h e ring Z in the c o m m u t i n g variables y a n d x l . Let us find the coefficient g-3 in t h e e x p r e s s i o n (4) for Olc. Direct calculations s h o w t h a t g-3 X l 2 y - 3 ( Y -1 x-1). --

--

On expressing t h e e l e m e n t (1 - 5 ~ ) O l u in the f o r m (4), we see t h a t in t h e g r o u p ring of t h e free Abelian g r o u p w i t h basis y , x l we h a v e t h e relation

y-1 _

= (1 -

y)

(5)

for s o m e g ( x l , Y) 6 Z [ ~ 1 , y • However, each of the elements of t h e ring Z [ ~ 1 , y• can u n i q u e l y be w r i t t e n in t h e f o r m ~ f j y J , w h e r e f j 6 Z[5~1]. On r e p r e s e n t i n g the e l e m e n t g ( ~ l , y) in this form, we see t h a t r e l a t i o n (5) is impossible. T h e resulting c o n t r a d i c t i o n proves t h e t h e o r e m . [] P r o p o s i t i o n 1 a n d t h e t h e o r e m i m p l y the following assertion. Corollary.

/ n the group Fr(~cY~2) we can choose two nonconjugate elements with equal n o r m a l clo-

sures.

A similar t h e o r e m for t h e g r o u p s Fr(2[ 2) was proved by S h m e l ' k i n in [8]. References 1. M . J . Evans, "Presentations of the free metabelian group of rank 2," Canad. Math. Bull., 37, No. 4, 468-472 (1994). 2. C. K. Gupta, N. D. Gupta and G. A. Noskov, "Some applications of Artamonov-Quillen-Suslin theorems to metabelian inner rank and primitivity," Canad. J. Math., 46, No. 2, 298-307 (1994). 3. N. S. Romanovskii, "Free subgroups of finitely-presented groups," Algebra i Logika [Algebra and Logic], 16, No. 1, 88-97

(1977). 4. N. S. Romanovskii, ~A generalized theorem on freedom for pro-p-groups," Sibirsk. Mat. Zh. [Siberian Math. J.], 27, 154-170 (1986). 5. C. K. Gupta and N. S. Romanovskii, "A generalized Freiheitssatz for centre-by-metabelian groups," Bull. London Math.

Soc., 24, 71-75 (1992). 6. G. G. Yabanzhi, "On groups that are finitely presented in the varieties P2~2 and fTl2~ ," Algebra i Logika [Algebra and Logic], 20, No. 1, 109-120 (1981). 7. C. K. Gupta and N. S. Romanovskii, "A generalized Freiheitssatz for the variety P2~ A ff[2P2," Algebra Colloq., 1, 193-200 (1994). 8. A. L. Shmel'kin, "Two remarks on free solvable groups," Algebra i Logika [Algebra and Logic], 6, No. 2, 95-109 (1967). 9. E, I. Timoshenko, "On the inclusion of given elements in a basis of a free metabelian group," Dep. VINITI, No. 2699-B88. 10. V. A. Roman'kov, "Criteria for the primitivity of a system of elements of a free metabelian group," Ukrain. Mat. Zh. [Ukrainian Math. J.], 43, No. 7-8, 996-1002 (1990). 11. E. I. Timoshenko, "Algorithmic solvability of the inclusion problem in a basis for a free metabelian group," Mat. Zametki [Math. Notes], S1, No. 3, 117-121 (1992). 12. N. S. Romanovskii, "A theorem on freeness for groups with one defining relation in varieties of solvable and nilpotent groups of given degrees," Mat. Sb. [Math. USSR-Sb.], 89, No. 1, 93-99 (1972). 13. N. S. Romanovskii, "Certain algorithmic problems for solvable groups," Algebra i Logika [Algebra and Logic], 13, No. 1, 26-34 (1974). NOVOSIBIRSK STATE BUILDING ACADEMY

Translated by A. I. Shtern 743