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On the Fitting Subgroup of a Polycylic-By-Finite Group and Its Applications

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On the Fitting Subgroup of a Polycylic-By-Finite Group and Its Applications Journal of Algebra 242, 176᎐187Ž. 2001 doi:10.1006rjabr.2001.8803, available online at http:rrwww.idealibrary.com on On the Fitting Subgroup of a Polycyclic-by-Finite Group and Its Applications Bettina Eick View metadata, citation and similar papers at core.ac.uk brought to you by CORE ¨ Fachbereich fur¨¨ Mathematik und Informatik, Uni ersitat Kassel, provided by Elsevier - Publisher Connector 34109 Kassel, Germany E-mail: [email protected] Communicated by Walter Feit Received May 12, 2000 We present an algorithm for determining the Fitting subgroup of a polycyclic- by-finite group. As applications we describe methods for calculating the centre and the FC-centre and for exhibiting the nilpotent-by-abelian-by-finite structure of a polycyclic-by-finite group. ᮊ 2001 Academic Press 1. INTRODUCTION The Fitting subgroup of a polycyclic-by-finite group can be defined as its maximal nilpotent normal subgroup. For Fitting subgroups of finite groups various other characterizations are known: For example, they can be described as the centralizer of a chief series. We observe that the Fitting subgroup of a polycyclic-by-finite group can also be characterized as the centralizer of a certain type of series, and this series can be considered as a ‘‘generalized chief series.’’ We describe a practical algorithm for computing a generalized chief series of a polycyclic-by-finite group G. Our method is based on the determination of radicals of finite-dimensional KG-modules, where K is either a finite field or the rational numbers. Once a generalized chief series of G is given, we can construct its centralizer and thus we obtain an algorithm to compute the Fitting subgroup of G. The determination of the Fitting subgroup has a number of applications. We show how to use the Fitting subgroup to construct the centre and the 176 0021-8693r01 $35.00 Copyright ᮊ 2001 by Academic Press All rights of reproduction in any form reserved. FITTING SUBGROUPS 177 FC-centre of a polycyclic-by-finite group. Further, a well-known theorem of Mal’cev asserts that a polycyclic-by-finite group is nilpotent-by-abelian- by-finite, and we describe a method to exhibit this structure. The algorithmic theory of polycyclic-by-finite groups has been investi- gated inwx 1 and it is observed there that the Fitting subgroupŽ and various other interesting subgroups. of a polycyclic-by-finite group can be deter- mined algorithmically. However, the approach used for this purpose is an algorithm in the classical sense and has not been invented for practical applications. The algorithms introduced here have been implemented for polycyclic groups in the Polycyclic packagewx 7 of the computer algebra system GAP wx15 . This implementation shows that the described methods are effective and thus we obtain a first practical approach to determine the Fitting subgroup, the centre, and the FC-centre of an infinite polycyclic group. 2. A CHARACTERIZATION OF FITTING SUBGROUPS Let N F M F G be normal subgroups of a group G.If NrM is an r ކ elementary abelian p-group, then N M is an pG-module under the natural conjugation action of G.If NrM is free abelian, then NrM is a ޚG-module and, by extending the coefficient ring, we can also consider NrM as a ޑG-module in this case. Recall that a module is called semisimple if it is a direct sum of irreducible modules. THEOREM 2.1. Let G be a polycyclic-by-finite group and let P be its largest s d d иии d d s polycyclic normal subgroup. Let P P12P PnnP q1 1 be a normal series with elementary or free abelian factors such that each free abelian factor is semisimple as a ޑP-module and each elementary abelian ކ p-factor is semisimple as an p P-module. Then n s r FitŽ.G F CPPii ŽP q1 .. is1 Proof. First note that FitŽ.G F P and thus that FitŽ.G s Fit Ž.P . s Fn r l l Let C is1 CPPiŽ.P iq11. Then the series C P ,...,C Pnq1 is a central series of C, and hence we obtain that C is nilpotent. Clearly, Ceᎏ P and thus C F FitŽ.P . It remains to show that FitŽ.P F C; that is, each nilpotent normal subgroup F of P centralizes the given series. First recall that for any wx wx abelian Aeᎏ P we have A, F - A: either, if A Fu F, then A, F F A l F - A or, if A F F, then intersecting a central series of F with A yields an F-central series of A, and hence wxA, F - A. 178 BETTINA EICK s r Now let A PiiP q1 be a factor of the given series of P. We can s assume that Piq1 1 and A is a subgroup of P. By assumption, A is s ކ s ޑ s semisimple as a KP-module for K p or K and thus A A1 = иии = e s Arjwith KP-irreducible subgroups A ᎏ P. We consider B Aj for some j.If B is elementary abelian, then wxB, F - B as observed above wx ކ and B, F is an p P-submodule of B. Since B is irreducible, we obtain wxB, F s 1 and F centralizes B.If B is free abelian, then F centralizes a series in the finite abelian group BrB p for every prime p by the above argument. Hence, if d is the rank of B, then the d-fold commutator wxF p wxs BF,d B . Thus BF,d 1 and F centralizes a finite series in B.In ) particular, CFBBŽ. 1. Further, CFŽ.is normal in P and thus, since B is ޑ w x - ϱ P-irreducible, B : CFBŽ. But centralizers in free abelian groups s are pure subgroups and thus we obtain CFBŽ. B, as desired. Clearly, we could replace ‘‘semisimple’’ with ‘‘irreducible’’ in Theorem 2.1. For finite groups this reduces then to the well-known characterization of the Fitting subgroup as the centralizer of a chief series. However, there is no practical algorithm known so far for determining a series with irreducible factors in an infinite polycyclic group, while a semisimple series can be constructed as outlined in Section 3. 3. DETERMINING THE FITTING SUBGROUP We use Theorem 2.1 to determine the Fitting subgroup of a polycyclic- by-finite group G. We assume that the largest polycyclic normal subgroup P of G is given explicitly and we can compute effectively with P: for example, a polycyclic presentation for P is suitable for our approaches; see Section 8 for details. We construct FitŽ.G in three steps. Step 1. We determine a normal series with elementary or free abelian factors of P. For example, we can use the effective methods inwx 5 to construct a computationally useful series of this type, that is, a series with small sum of the ranks of its factors. s ކ Step 2. Each factor in this initial series is a KP-module for K p or K s ޑ. We refine each factor by a submodule series with semisimple factors and thus we obtain a generalized chief series of P. s d иии d s Step 3. If P P1 Pnq1 1 is the determined generalized chief s d иии d series, then we construct the sequence of centralizers P C1 ᎏᎏCnq1 defined by C [ CPŽ.y rP . This yields Fit Ž.G s C q . iCiiy1 1 in1 For Step 2 we consider the radical Rad KPŽ.V of a finite-dimensional KP-module V, that is, the intersection of its maximal submodules. Since FITTING SUBGROUPS 179 - r Rad KPŽ.V V and V RadKP Ž.V is semisimple, we obtain that the radical s ކ series of V yields a series with semisimple factors. If K p, then practical methods of determining radicals are well-known; see for example wx14 . The case K s ޑ is considered in Section 3.2. In Step 3 we have to construct centralizers of semisimple KP-modules V. ކ Again, if V is an p P-module, then such methods are well-known. We discuss the case of a ޑP-module V in Section 3.3. Remark 3.1. Methods which are similar to the approaches described in this section have been used by Ostheimerwx 11 to exhibit the structure of a polycyclic-by-finite rational matrix group. 3.1. Congruence Subgroups Let H F GLŽ. d, ޑ be a finitely generated group. Then there exists a set of primes ␲ such that H F GLŽ. d, ޑ␲ , where ޑ␲ is the subring of ޑ of a ␲ those rationalsb with b divisible by primes in only. Consider a prime p f ␲ ޑ ޚ with p . Then we can embed ␲ into the p-adic numbers p. Combin- ޚ ª ޚ r ޚ s ކ ing this embedding with the natural homomorphism ppppp , ␫ ޑ ª ކ we obtain a ring homomorphism : ␲ p. This homomorphism extends naturally to the p-congruence homomorphism ␫ ␺ : H ª GL d, p : h ¬ h . piŽ.Ž., jii, j ž/ Ž., j i, j ␺ The kernel Hppof is called the p-congruence subgroup of H. The following fundamental theorem is proved inwx 4; Lemma 9 . THEOREM 3.1Ž. Dixon . Let ␲ be a set of primes and H F GL Ž. d, ޑ␲ be polycyclic-by-finite. If p is an odd prime with p f ␲ , then H is torsion-free X p and Hp is unipotent. There are various possible approaches to determine generators or normal subgroup generators for Hp.
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