Minimal Normal Subgroups of Dinilpotent Groups

Journal of Algebra 234, 480᎐491 Ž2000. doi:10.1006rjabr.2000.8551, available online at http:rrwww.idealibrary.com on

Minimal Normal Subgroups of Dinilpotent Groups Derek J. S. Robinson1 Department of Mathematics, Uni¨ ersity of Illinois, Urbana, Illinois 61801 E-mail: [email protected]

and Stewart E. Stonehewer Mathematics Institute, Uni¨ ersity of Warwick, Co¨ entry CV4 7AL, United Kingdom E-mail: [email protected] Communicated by Gernot Stroth Received April 3, 2000 PROF. DR. HELMUT WIELANDT, ZUM

90TEN

GEBURTSTAG GEWIDMET

If a finite group G is the product of two nilpotent subgroups A and B and if N is a minimal normal subgroup of G, then AN or BN is nilpotent. This result is extended to several classes of infinite groups. 䊚 2000 Academic Press Key Words: nilpotent subgroups; factorizations; minimal normal subgroups; chief factors.

1. INTRODUCTION A group G is called dinilpotent if it has a nilpotent factorization; that is, G s AB where A and B are nilpotent subgroups. Such groups have been widely studied over the past 50 years: for an overview of the subject the reader may consult the book by Amberg et al. w1x. Here we are concerned with the following question: Ž*. If N is a minimal normal subgroup of G, is it true that either AN or BN is nilpotent? 1 This research was carried out during the summer of 1999, when the first author was a visitor at the University of Warwick, supported by a grant from EPSRC.

480 0021-8693r00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.

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More generally the problem could be stated for chief factors of G: if NrL is a chief factor of G, is either ANrL or BNrL nilpotent? The first result in this direction was found by the present authors in w11x, where it was shown that if A and B are abelianᎏso that G is diabelianᎏthen a minimal normal subgroup N of G is centralized by A or B. Thus AN or BN is abelian since G is metabelian by a well-known theorem of Ito ˆ w4x. Shortly thereafter Stonehewer w16x was able to prove that Ž*. is true when G is a finite group. In this case neither A nor B need centralize N, as a factorization of S 4 shows. However, if G is nilpotentby-abelian, the nilpotence of AN does imply that w A, N x s 1 Žsee ŽII. in Section 3.. Of course Stonehewer’s proof makes use of the celebrated Wielandt᎐Kegel theorem: a finite group is soluble if it is the product of two nilpotent subgroups. Since nothing is known about the structure of infinite dinilpotent groups in general, most investigations have been concerned with soluble dinilpotent groups. However, problem Ž*. seems untouchable even for soluble groups. Indeed almost all results obtained so far for infinite soluble dinilpotent groups have required a finiteness condition to be imposed on the group G. In a recent M.Phil. dissertation w14x, Smith proved that the question Ž*. has a positive answer if G is nilpotent-by-finitely generated nilpotent and A l B s 1. Smith also gave a simplified proof of Stonehewer’s theorem. Finally we mention an article by Franciosi et al. w3x, in which the range of the investigation is extended by showing that if G s AB is a periodic radical group and A and B are locally nilpotent, one of which is hyperabelian, then AN or BN is locally nilpotent. This result provides evidence that the presence of elements of infinite order is a complicating factor in the problem.

2. RESULTS Our first result has the virtue of requiring no finiteness restrictions on the group. THEOREM 1. Let G be a metabelian group with a nilpotent factorization G s AB. Then e¨ ery chief factor of G is centralized by either A or B. As a nice consequence we record: COROLLARY. Suppose that the metabelian group G has a triple nilpotent factorization G s AB s BC s CA, with A, B, and C nilpotent subgroups. Then e¨ ery chief factor of G is central. Thus G is a Z-group in the sense of Kurosh w8x.

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The corollary follows at once, since two of A, B, and C must centralize a chief factor. For comparison we recall a result of Kegel w6x: a finite group is nilpotent if it has a triple nilpotent factorization. Indeed the conclusion is valid for any finitely generated soluble-by-finite group with a triple factorization, since all its finite quotients are nilpotent Žsee w10, p. 477x.. Finally, Franciosi and de Giovanni w2x established the nilpotence of soluble groups of finite total rank Ži.e., S1-groups., with a triple nilpotent factorization. All our subsequent results are for nilpotent-by-polycyclic groups. Notice that in such groups the chief factors are necessarily finite, by Roseblade’s theorem w12x that simple modules over polycyclic groups are finite. The most difficult result to establish is the following generalization of Smith’s theorem. THEOREM 2. Let G be a nilpotent-by-polycyclic group with a nilpotent factorization G s AB such that A l B is periodic. If N is a minimal normal subgroup of G, then AN or BN is nilpotent. While the problem Ž*. remains open for general nilpotent-by-polycyclic groups, we have been able to settle it for some significant subclasses. THEOREM 3. Let G be an abelian-by-Ž finitely generated nilpotent . group with a nilpotent factorization G s AB. If N is a minimal normal subgroup of G, then either AN or BN is nilpotent. THEOREM 4. Let G be a Ž nilpotent of class F 2.-by-Ž finitely generated abelian. group, with a nilpotent factorization G s AB. Then a chief factor of G is centralized by A or B. This result leads to a triple nilpotent factorization theorem in the same way as the Corollary of Theorem 1. THEOREM 5. Let G be a finitely generated Ž nilpotent of class F 3.-byabelian group, with a nilpotent factorization G s AB. Then a chief factor of G is centralized by A or B. The last three results all depend on Theorem 2. Clearly our results are partial in nature, but it should be stressed that the problem Ž*. is an exceptionally challenging one. No doubt refinements in the techniques of the present work, together with new ideas, will serve to establish the truth of Ž*. for wider classes of groups.

3. THE METABELIAN AND OTHER SIMPLE CASES We begin with some useful observations of a general character. Let G be a group with a nilpotent factorization G s AB, and let N be a minimal

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normal subgroup of G. Using Zorn’s lemma, we can choose a normal subgroup L which is maximal subject to N l L s 1. Then N (G NLrL and NLrL is contained in every non-trivial normal subgroup of GrL. Thus GrL is a monolithic group. If we can prove our theorem for GrL, then it follows immediately that AN or BN is nilpotent. Therefore we have an important reduction in the problem of establishing Ž*.. ŽI. If the hypotheses on G are inherited by quotients, then it can be assumed that N lies in e¨ ery non-tri¨ ial normal subgroup of G. As a first application we prove LEMMA 1. Let N be a minimal normal subgroup of a finitely generated group G and suppose that G is abelian-by-polycyclic. If G s AB is a nilpotent factorization, then AN or BN is nilpotent. Proof. As was pointed out above, N is finite. G and its quotients are also residually finite, by the theorem of Jategaonkar w5x and Roseblade w13x. Therefore by ŽI. we can assume G to be finite, when the result follows from Stonehewer’s theorem w16x. A second useful reduction is ŽII.

If G is nilpotent-by-abelian and AN is nilpotent, then w N, A x s 1.

Here the notation of Ž*. is being used. Proof of ŽII.. Since G⬘ is nilpotent and N is minimal normal in G, we have w N, G⬘x s 1. Hence w N, A x s w N, AG⬘x 1 G. If w N, A x / 1, then N s w N, A x s w N, AN x. But then AN cannot be nilpotent. Proof of Theorem 1. Since G is metabelian, we can pass to a quotient group and assume that N is contained in every non-trivial normal subgroup, as indicated in ŽI.. Suppose that A l G⬘ / 1; then N F Ž A l G⬘. G . Now for large enough i we have G G G Ž A l G⬘ . , i A s Ž A l G⬘ . , i AG⬘ s w A l G⬘, i AG⬘ x

s w A l G⬘, i A x s 1. G

Therefore AN is nilpotent, and this shows that w N, A x s 1. Thus we may assume that A l G⬘ s 1 s B l G⬘. But now A and B are abelian, and the result follows directly from w11x. Two more results of a simple character are recorded next. PROPOSITION 1. Let G be a group with a nilpotent factorization G s AB, and let N be a minimal normal subgroup of G. Then AN or BN is nilpotent if

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one of the following conditions holds: Ži. Žii.

G is a finitely generated linear group. G is abelian-by-finite.

Proof. Ži. By a well-known theorem of Mal’cev, G is residually linear of degree n over finite fields Žsee w17, p. 51x.. In particular G is residually finite and so there is a normal subgroup M of finite index in G such that N l M s 1. But then N (G NMrM and we can appeal to the finite case. Žii. There is a normal abelian subgroup M of finite index in G. By ŽI. we can assume that N is contained in every non-trivial normal subgroup of G. Suppose that M has an element a of infinite order. Then aG is a finitely generated infinite abelian group and NF

F Ž aG .

m

s 1.

m)0

By this contradiction M and hence G are periodic. Now apply the theorem of Franciosi et al. w3x mentioned above to deduce that AN or BN is locally nilpotent. Since N is finite, it follows that AN or BN is nilpotent.

4. PROOFS OF THEOREMS 2 AND 3 If N is a normal subgroup of a factorized group G s AB, the factorizer of N is the subgroup X Ž N . s AN l BN s Ž A l BN . N s Ž AN l B . N s Ž A l BN . Ž AN l B . . If X Ž N . s N, then N is said to be factorized; in this case N s Ž A l N .Ž B l N .. Our first object is to establish the following unpublished result of Smith w14x. PROPOSITION 2. Let G be a nilpotent-by-polycyclic group with a nilpotent factorization G s AB. Then the Fitting subgroup of G is factorized. First we need to prove LEMMA 2. Let G be a group with a triple nilpotent factorization G s AB s AL s BL where L is normal in G. If GrL is finitely generated, then G is nilpotent.

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Proof. By Hall’s nilpotency criterion, it is enough to show that GrL⬘ is nilpotent. So we may assume that L is abelian. Then

w A l L, i G x s w A l L, i AL x s w A l L, i A x s 1 for large enough i. Thus A l L and also B l L lie in the hypercentre, which allows us to assume that A l L s 1 s B l L. Hence A and B are finitely generated, as is G. Since finite quotients of G are nilpotent by Kegel’s theorem w6x, G is nilpotent, by w10, p. 477x. Proof of Proposition 2. Let F denote the Fitting subgroup of G; certainly F is nilpotent. Put X s X Ž F .. It is sufficient to show that X is subnormal and nilpotent. Now X s Ž A l BF .Ž AF l B . s Ž A l BF . F s Ž AF l B . F, so that Lemma 2 can be applied to show that X is nilpotent. Next X s AF l BF, so XrF is subnormal in AFrF and in BFrF. Let ␪ be any homomorphism from GrF into a finite group. Then Ž XrF . ␪ is subnormal in Ž AFrF . ␪ and in Ž BFrF . ␪ . By a theorem of Wielandt w18x, Ž XrF . ␪ is subnormal in Ž GrF . ␪. Since GrF is polycyclic, a theorem of Kegel w7x shows that XrF is subnormal in GrF. We shall need one further result. LEMMA 3. Let G be a group with a nilpotent factorization G s AB. Assume that Op Ž G . is locally finite and GrOp Ž G . is finite and soluble. Then A p⬘ Bp⬘ is a Hall p⬘-subgroup of G. Proof. Let L s ² A p⬘ , Bp⬘ :, which is a finite soluble group. Let H be a Hall p⬘-subgroup of L containing A p⬘. Then Bp⬘ F H ab for some a in A and b in B, and hence Bp⬘ F H a. Also A p⬘ s Ž A p⬘ . a F H a. Therefore L is a p⬘-group and L l Op Ž G . s 1. By a result of Pennington w9x, A p⬘ Bp⬘ Op Ž G .rOp Ž G . is a Hall p⬘-subgroup of GrOp Ž G ., and it contains LOp Ž G .rOp Ž G .. Hence L s L l Ž A p⬘ Bp⬘ Op Ž G . . s A p⬘ Bp⬘ . If L1 is a finite subgroup containing L, then < L1Op Ž G . : LOp Ž G .< is a power of p, as < L1 : L < must be. Thus L is a Hall p⬘-subgroup. Proof of Theorem 2. First note that N is a finite elementary abelian p-group for some prime p, since G is nilpotent-by-polycyclic. Let F be the Fitting subgroup of G. Then N F F and F is nilpotent; hence w N, F x s 1. Note that A l B is subnormal in A and in B, so it is subnormal in G, by a result of Stonehewer w15x. It follows that A l B F F. Now let T be a periodic normal subgroup of G contained in ZŽ F ., the centre of F. We claim that AT l BT is periodic. Indeed let x g AT l BT and write x s at1 s bt 2 ,

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where a g A, b g B, and t i g T. Since AT l BT F X Ž F . s F by Proposition 2, we obtain x g F. It follows that w a, t 1 x s 1 s w b, t 2 x, and thus x m s a m s b m g A l B for some m ) 0. Therefore x has finite order. This observation allows us to factor out by ZŽ F . p⬘ , which may have the effect of enlarging the Fitting subgroup. However, GrF is polycyclic, so after a finite number of such reductions we reach a situation where Fp⬘ s 1. At this point we introduce a critical pair of subgroups. Since N is finite and N F F s Ž A l F.Ž B l F., we can write N s  a i bi N i s 1, 2, . . . , m4 , with a i in A l F and bi in B l F. Define A1 s ² a1 , a2 , . . . , a m :

and

B1 s ² b1 , b 2 , . . . , bm : .

Then V [ A1 N s B1 N s A1 B1. Since w N, F x s 1, we have 1 s w a i bi , a i x s w bi , a i x. Therefore 1 s Ž a i bi . p s a ip bip and a ip s byp g A l B. It foli lows that a i and bi have finite order, necessarily a power of p since Fp⬘ s 1. Thus A1 and B1 are finite p-groups, as V must be. Let X s X Ž N ., the factorizer of N in G. Then X s Ž A l BN . N s Ž AN l B . N s Ž A l BN . Ž AN l B . . Now A l BN s A l A1 B s A1Ž A l B . and X s A1 N Ž A l B ., showing that < X : A l B < is a power of p. Since X F X Ž F . s F, we conclude that X is a nilpotent p-group. Let p e denote the exponent of V and put W s ⍀ e Ž X .. Then W has finite exponent a power of p, since X is nilpotent. Note that V F W, since V F X. We show that W is factorized. In the first place N F A1 B1 : ŽW l A.ŽW l B .. Since W F X s AN l BN, it follows that W s W l X s W l Ž Ž A l BN . N . s Ž W l A l BN . N : Ž W l A. Ž W l B . . Thus W s ŽW l A.ŽW l B .. Now put J s NG ŽW . and C s CG ŽW .. The next step is to show that J s Ž J l A. Ž J l B .

and

JrC is p-by-finite.

To establish this, let ab g J, with a g A and b g B. Then WasWb

y1

F AN l BN s X .

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Therefore W a F ⍀ e Ž X . s W and ŽW l A. a F W l A. Now W l A has finite exponent a power of p and A is nilpotent; hence it is easy to see r that w W l A, A p x s 1 for some r ) 0. Then pr

a

W l A s Ž W l A . a F Ž W l A . F W l A, so that ŽW l A. a s W l A. Since W F AN, we see that W s ŽW l A. N and so W a s W; i.e., a g J l A. Similarly b g J l B and thus J s Ž J l A.Ž J l B .. r Next A p centralizes W l A. Therefore since W s ŽW l A. N, we have r r A p F J and also B p F J. Put C0 s CG Ž N .. Then GrC0 is finite and r r ArŽ C0 l A p . is p-by-finite. But C0 l A p F C, because W s ŽW l A. N. Therefore Ž J l A.CrC is p-by-finite, as is Ž J l B .CrC. Thus the polycyclic group JrŽ J l F .C is the product of two periodic subgroups Ž J l A.Ž J l F .CrŽ J l F .C and Ž J l B .Ž J l F .CrŽ J l F .C. Therefore JrŽ J r l F .C is finite. Also F p F C for large enough r, so that JrC is p-by-finite. Note that JrC has finite exponent. Next let Q be a Hall p⬘-subgroup of J s JrC. By Lemma 3 we can assume that k

Q s Ap Bp

k

for some k, where A s ACrC and B s BCrC. Here it is relevant that A r r and B have finite exponent and A p , B p F J, so that A p⬘ , Bp⬘ F J. Consider the semidirect product K s Q h W, where Q acts on W via conjugation. Thus K is a periodic soluble group. Furthermore W s ŽW l A.ŽW l B . implies that k

k

K s Ž A p h Ž W l A . .Ž B p h Ž W l B . . , and this is a nilpotent factorization. By the theorem of Franciosi et al. w3x, the problem Ž*. has a positive solution for K. Let M be a minimal normal subgroup of K contained in N. Then either k k k k A p ŽW l A. M or B p ŽW l B . M is nilpotent. But A p and B p are k k p⬘-groups, so this means that w M, A p x s 1 or w M, B p x s 1. Denote by NA the subgroup generated by all the minimal K-invariant subgroups of N k which are centralized by A p , with the corresponding definition of NB . Since N is completely reducible as a Q-module, N s NA NB . Evidently NA is A-invariant and NB is B-invariant. In addition ANA and BNB are nilpotent, since ArCAŽ NA . and BrCB Ž NB . are finite p-groups.

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Finally recall that C0 s CG Ž N .. Now we consider the semidirect product U s Ž GrC0 . h N. This is a finite group with a nilpotent factorization U s Ž A* h NA .Ž B* h NB ., where A* s AC0rC0 and B* s BC0rC0 . Therefore AN or BN is nilpotent by the finite case. This completes the proof of Theorem 2. Proof of Theorem 3. By hypothesis G has an abelian normal subgroup L such that GrL is finitely generated and nilpotent. Let F denote the Fitting subgroup of G, so that L F F. As usual we may assume that N is contained in every non-trivial normal subgroup of G. First we show that Ži. E¨ ery abelian normal subgroup containing L is periodic. Suppose that H is such a subgroup and x g H. Then x G is a finitely generated GrH-module and GrH is finitely generated and nilpotent. By Hall’s theorem Žsee w10, p. 470x. x G is residually finite as a GrH-module. Since N is finite, it follows that x G is finite. Thus H is periodic. Žii. CG Ž L.rL is finite. If this is false, CG Ž L.rL l ZŽ GrL. has an element of infinite order, say zL. But then ² z, L: is a non-periodic abelian normal subgroup of G, in contradiction to Ži.. Žiii. The proof is completed by an induction on the Hirsch length of GrCG Ž L.; this can be assumed positive, since otherwise G is periodic and Theorem 2 applies. Suppose first that FrL is finite; then F is periodic. Now A l B is subnormal in A and in B and hence in G by w15x. Therefore A l B F F and again Theorem 2 can be applied to give the result. We may therefore assume that FrL is infinite, from which it follows that FrL l ZŽ GrL. contains an element zL of infinite order. A G-endomorphism ␪ of L is defined by x ␪ s w x, z x. Since ² z, L: is nilpotent, there is a positive integer k such that ␪ k / 0 s ␪ kq1. Now ImŽ ␪ k . 1 G and N F ImŽ ␪ k . (G LrKerŽ ␪ k .. Since z centralizes LrKerŽ ␪ k ., we have ² z, CG Ž L . : F CG Ž LrKer Ž ␪ k . . s D, say. In addition ² z : l CG Ž L. s 1, by Žii.. Therefore Ž GrKerŽ ␪ k ..r Ž DrKerŽ ␪ k .. has smaller Hirsch length than GrCG Ž L., and the theorem is true for this group. It follows that AN or BN is nilpotent.

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5. PROOFS OF THEOREMS 4 AND 5 The proofs of our last two results rest on the following technical lemma. LEMMA 4. Let G be a group with a nilpotent factorization G s AB. Assume that there is a normal nilpotent subgroup F such that GrF is abelian. Define Z s ZŽ F . and G

A s AŽ A l Z . ,

G

B s BŽ B l Z. .

Then: Ži. A l Z and B l Z are normal subgroups of G; A and B are nilpotent and G s AB. Žii. If A l B l Z l ZŽ G . s 1, then A l B l Z s 1. Proof. In the first place G G G Ž A l Z . , r A s Ž A l Z . , r AF s w A l Z, r AF x

s w A l Z, r A x s 1 G

for sufficiently large r. Hence A is nilpotent, and similarly B is nilpotent. Clearly G s AB. Also A l Z s Ž A l Z . G , so A l Z 1 G, and also B l Z eG. Next put D 1 s A l B l Z, and assume that D 1 / 1. Now D 1 1 G, since D 1 s Ž A l Z . l Ž B l Z ., and hence D 1 l ZŽ A. / 1, since A is nilpotent. Put D 2 s Ž D 1 l ZŽ A.. G . Then D 2 F Z and D 2 , A s D 2 , AF s D 1 l Z Ž A . , AF

G

s D1 l Z Ž A . , A

G

s 1.

Now D 2 F Ž B l Z . G s B l Z, so that D 2 l ZŽ B . / 1, since B is nilpotent. Finally D 2 l ZŽ B . F A l B l Z l ZŽ G .. Proof of Theorem 4. By hypothesis G has a nilpotent normal subgroup F with class at most 2 and GrF finitely generated abelian. Let N be a minimal normal subgroup of G. Then it suffices to show that AN or BN is nilpotent. We know that N is a finite elementary abelian p-group for some prime p. As usual we can suppose N to lie inside every non-trivial normal subgroup of G. Put Z s ZŽ F .. If z g Z has infinite order, then z G is a finitely generated GrF-module and its torsion subgroup has finite exponent, so that Ž z G . k is torsion-free for some k ) 0, which is impossible. Thus Z is periodic, and clearly it must be a p-group. According to Lemma 4, AŽ A l Z . G and B Ž B l Z . G are nilpotent. From this we may infer that A l Z s 1 s B l Z, since otherwise N is contained in Ž A l Z . G or Ž B l Z . G and AN or BN is nilpotent.

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Suppose next that Ž FrZ . l ZŽ GrZ . is non-trivial and let zZ be a non-identity element in the intersection. The assignment xF⬘ ¬ w x, z x is a G-module homomorphism ␪ from FrF⬘ to Z. Thus 1 / ImŽ ␪ . s w F, z x 1 G, from which it follows that G

N F Im Ž ␪ . ( Ž FrF⬘ . rKer Ž ␪ . . But GrF⬘ is metabelian. Thus Theorem 1 shows that w A, N x or w B, N x is trivial. So we can suppose that Ž FrZ . l ZŽ GrZ . s 1. Now apply Lemma 4Žii. to the group GrZ. It follows that AZ l BZ l F s Z; that is, X l F s Z, where X s X Ž Z .. Hence X l A ( Ž X l A. ZrZ F XrZ ( XFrF. Therefore X l A is finitely generated abelian, and of course the same is true of X l B. But X s Ž X l A.Ž X l B ., so X is polycyclic, by a theorem of Lennox and Roseblade Žsee w1, p. 8x.. It now follows that Z is finite and hence F has finite exponent. Finally, let F1 denote the Fitting subgroup of G. Then F1rF is finitely generated abelian and ZŽ F1 . is periodic, by the previous argument. Hence ZŽ F1 . has finite exponent, as does F1. However, A l B F F1 , so we are back in the situation of Theorem 2. Proof of Theorem 5. By hypothesis G is finitely generated and has a normal subgroup F which is nilpotent of class F 3, with GrF abelian. As usual the minimal normal subgroup N is a finite elementary abelian p-group and we can assume that it is contained in every non-trivial normal subgroup of G. Put Zi s Zi Ž F .; then Z1 is a p-group by the usual argument. Notice also that we can take A l Z1 and B l Z1 to be trivial, as in the last proof. Suppose first that zZ1 is a non-trivial element of Ž FrZ1 . l ZŽ GrZ1 .. The assignment xF⬘ ¬ w x, z x is a G-module homomorphism ␪ from FrF⬘ to Z1 , and G

N F Im Ž ␪ . ( Ž FrF⬘ . rKer Ž ␪ . . Since Ž GrF⬘.rKerŽ ␪ . is Žnilpotent of class F 2.-by-abelian, we can deduce the result from Theorem 4. So we may assume that Ž FrZ1 . l ZŽ GrZ1 . s 1. Applying Lemma 4 to GrZ1 , we conclude that X Ž Z1 . l Z2 s Z1 . Since GrZ2 is a finitely generated metabelian group, FrZ2 is a noetherian GrF-module. Consequently there is a G-invariant subgroup H such that Z2 F H F F, w H, r G x F Z2 for some r ) 0, and ZŽ GrH . l Ž FrH . s 1. Hence HrZ2 is finitely generated abelian. Apply Lemma 4 once more, this time to the group GrH. We conclude that X Ž H . l F s H,

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491

from which it follows that X Ž H .rZ2 is polycyclic. Therefore X Ž Z1 .rZ1 is polycyclic, since X Ž Z1 . l Z2 s Z1. Furthermore A l X Ž Z1 . ( Ž A l X Ž Z1 . . Z1rZ1 F X Ž Z1 . rZ1 , and A l X Ž Z1 . is polycyclic, as is B l X Ž Z1 . by the same reasoning. It follows that X Ž Z1 . s Ž A l X Ž Z1 . .Ž B l X Ž Z1 . . is polycyclic by the result of Lennox and Roseblade Žsee w1x.. Thus Z1 is finite. Now argue as in the previous proof that the Fitting subgroup is periodic. Then the result follows from Theorem 2.

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