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

Home , Chief series, Fitting subgroup, Polycyclic group

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

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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 determined 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. For example, one can compute a presentation for the finite image of H in GLŽ. d, p and evaluate the

relators of this presentation to obtain normal subgroup generators for Hp. In our later applications H will be given as the image of a polycyclically

presented group. In this case it is most effective to compute Hp by ކ d stabilizing each basis element in p under the action of H using the well-known stabilizer algorithm for finite polycyclic groups; see for exam- plewx 6 .

3.2. Radicals of ޑP-Modules Let P be a polycyclic group and V a d-dimensional ޑP-module with corresponding operation homomorphism ␥ : P ª GLŽ. d, ޑ . We denote 180 BETTINA EICK

P␥ s H F GLŽ. d, ޑ and let ޑwxH be the matrix algebra generated by the s elements of H. Clearly, Radޑ P Ž.V Radޑ H Ž.V .

THEOREM 3.2. Let H F GLŽ. d, ޑ␲ be polycyclic-by-finite, p an odd f ␲ s ޑ d prime with p , and Hp the p-congruence subgroup in H. Let V . s ޑwx Ž.a Radޑ H ŽV . V Radޑ H Ž H ..

Ž.b Radޑ Ž.V s Radޑ Ž.V . H H p X Ž.c If h g H , then VŽ. h y 1 : Radޑ Ž. V . p H p Ž.d Let H be abelian and B be a basis of ޑwxH . Then V is semisimple as a ޑH-module if and only if each element in B is diagonalizable o¨er ރ. Proof. Ž.a ޑwxH is finite dimensional and thus Artinian. wx- ϱ wx Ž.b H : Hp , and henceŽ. b follows by 16, 1.5 and 1.8 . ރ g X Ž.c By Theorem 3.1, Hppis triangularizable over and, if h H , y g then h is unipotent. Thus xhŽ.1 is nilpotent for each x Hp. Hence wx Ž.h y 1 g Radޑ Žޑ H .Ž., and c follows by Ž.Ž. a and b . Hpp Ž.d If each basis element of an abelian algebra is diagonalizable, then the basis elements are also simultaneously diagonalizable by a theorem of Schur. Thus V is semisimple in this case. The converse is obvious.

We compute the desired radical using Radޑ Ž.V s Radޑ Ž.V for a P H p suitable prime p by Theorem 3.2Ž. b . First, we compute a set of generators

Hppof H as described in Section 3.1. Then, by Theorem 3.2Ž. c , we obtain s wxy < g F W ²ŽVg, h 1. g, h Hp:Ž.Radޑ H V . We determine a basis for ޑ p the Hp-submodule R generated by W using a spinning algorithm. Now it remains to find Radޑ Ž.VrR s Radޑ Ž.V rR. Hence we pass to VrR. H ppH To simplify the notation we assume that R s 0.

By construction, Hp acts as an abelian group on V now and we want to use Theorem 3.2Ž. d to determine Radޑ H Ž.V . For this purpose we start to ޑwx p determine a basis B for Hp by using a spinning algorithm. We s л ޑwx initialize B and successively enlarge B until it spans Hp ; that is, ޑwx until each generator of Hp is contained in ²:B and ²:B is closed under multiplication with the generators of Hp. Whenever we add a new element h to B, we check that it is diagonalizable, as recalled in the following well-known lemma.

LEMMA 3.1. Let H F GLŽ. d, ޑ be abelian and h g ޑwxH . Let fŽ. x g ޑwx e1 иии er x , the minimal polynomial of h with factorization f1Ž. x frŽ. x into ¨ ޑ irreducible factors o er such that f1,..., fr are pairwise different. s иии s s Ž.a The matrix h is diagonalizable if and only if e1 er 1. / иии g ޑwx Ž.b If h is not diagonalizable, then 0 fh1Ž.fhr Ž.Radޑ H Ž H .. FITTING SUBGROUPS 181

g ޑwx Thus, if the considered element h Hp is not diagonalizable, then wx Lemma 3.1Ž. b yields an element k g Radޑ Žޑ H .. We obtain Vk F Hpp Radޑ Ž.V . Hence we construct the ޑH -module W generated by Vk and Hpp then we pass to VrW and recursively determine Radޑ Ž.VrW . H p Eventually, this approach yields a submodule W F Radޑ Ž. V with H p semisimple factor VrW by Theorem 3.2Ž. d . Then W s Radޑ Ž.V . H p The following theorem recalls another method which can be used to determine the radical of an arbitrary rational matrix algebra. This can be applied to obtain Radޑ P Ž.V as described in Theorem 3.2 Ž. a .

THEOREM 3.3Ž Dicksonwx 3. . Let A be a rational matrix algebra with basis s q иии q g a1,...,alij. Let M ŽŽTr a a ..1F i, jF l. Then e11 a eallRad A Ž.Aif s and only ifŽ. e1,...,eMl 0. Thus, once a basis of ޑwxH is known, the determination of a basis for ޑwx Radޑ H Ž H . reduces to solving a system of linear equations. But the dimension of ޑwxH can be as large as d 2 if H F GLŽ. d, ޑ , and hence the construction of a basis for ޑwxH may not be practical. But it is practical to use Dickson’s Theorem as an alternative to the method of Theorem 3.2Ž. d and Lemma 3.1 for determining the radical of an abelian algebra of the form ޑwxH . This alternative then avoids the construction and factorization of minimal polynomials.

3.3. Centralizers of Semisimple ޑP-Modules Let P be a polycyclic group and V s ޑ d a semisimple ޑP-module with the corresponding operation homomorphism ␥ : P ª GLŽ. d, ޑ␲ . Then s ␥ CVP Ž.ker Ž.. To compute this kernel, we use the following conse- quence of Theorems 3.1 and 3.2.

LEMMA 3.2. Let H F GLŽ. d, ޑ␲ be polycyclic-by-finite and suppose that the natural ޑH-module V is semisimple. If p is an odd prime with p f ␲ , then Hp is free abelian of finite rank. Let p be an odd prime with p f ␲ and consider the p-congruence ␺ ␥ homomorphism p. As a first step to computing kerŽ.we determine s ␥ и ␺ PppkerŽ., using the method of Section 3.1. By Lemma 3.2 we obtain that Pppacts as the free abelian group H on V. The kernel of this action can be obtained by constructing a set of defining relations for the given generators of Hp. In the following we consider two approaches to this problem. ޑwxޑ wx The elements of Hppare invertible elements of H . In turn, Hpis an abelian semisimple algebra and thus a direct sum of fields. This direct g ޑwxޑ wxs ޑ wx sum can be exhibited by computing c Hppwith H c as outlined inwx 4, Lemma 5 and then splitting V into irreducible c-submod- 182 BETTINA EICK

ޑwx ules. Thus we can reduce to the case that Hp is a field. Finding a set of defining relations for Hp is now a number theoretic problem. It can be solved using additive valuations for the given extension field of ޑ.An algorithm for this purpose is available in Kantwx 2 , and we refer the reader towx 12 for further information. Another approach toward finding a set of defining relations for genera- wx tors of Hp is described by Dixon 4 . There it is shown that the computation of defining relations for Hp is a finite problem; that is, it is sufficient e1 иии er to find those relations h1 hr in the generators h1,...,hrpof H with <

mation of the relations based on congruence homomorphisms for Hp using various different primes. We refer the reader towx 6 for further information.

4. DETERMINING THE CENTRE

Let G be a polycyclic-by-finite group. Then we can describe its centre ZGŽ.as the centralizer in Z ŽFit Ž..G under the action of G: s ZGŽ.CGZŽFitŽG.. Ž..

Thus to compute ZGŽ.we first determine FitŽ.G as in Section 3. The following well-known lemma can be used to construct ZŽŽ..Fit G by induction over a central series of the nilpotent group FitŽ.G .

LEMMA 4.1. Let G be a nilpotent group generated by g1,..., gr . Let F r s r ␯ r ª ¬ wx A ZŽ. G and C A ZGŽ.A and define ii: C A A: cA c, g . ␯ s Fr ␯ Then i is a homomorphism of abelian groups and ZŽ. G is1 kerŽ.i .

g g wxwxs c2 wxs Proof. For c12, c C and g G we have cc12, g c 1, gc 2, g wxwx ␯ c12, gc, g , since A is central. Thus each i is a well-defined homormor- phism. ␯ Since i in Lemma 4.1 is a homomorphism of abelian groups, we can determine its kernel by solving a system of linear homogeneous equations over A. Hence the centre of a nilpotent group can be determined effectively by an iterated application of Lemma 4.1. Once ZŽŽ..Fit G is determined, it remains to calculate its fixed points under the action of G. Since ZŽŽ..Fit G is abelian, we can consider it as a ޚG-module and obtain the fixed-points submodule using linear algebra: we successively determine the fixed points under each generator g of G and this, in turn, amounts to a null-space calculation for the action of g y 1. FITTING SUBGROUPS 183

5. DETERMINING THE FC-CENTRE

Let G be a group. An element g g G is called an FC-element in G if the conjugacy class g G is finite. The set of all FC-elements in G forms a characteristic subgroup: the FC-centre FCŽ. G . We recall some properties of FC-centers in the following lemma.

LEMMA 5.1. Let G be a polycyclic-by-finite group. Ž.a Ž Neumann . . The set of elements of finite order TŽ FC Ž G .. of FCŽ. G is a characteristic subgroup of FCŽ. G and FC Ž. G Ј F TFCG Ž Ž... Ž.b Let TN Ž G . be the largest finite normal subgroup in G. Then we obtain TŽ FC Ž G ..s TN Ž G . and FC Ž G .rTFCG Ž Ž ..s FC Ž GrTN Ž G ... Proof. ForŽ. a seewx 13, p. 121 , andŽ. b is straightforward. If G is a polycyclic group given by a polycyclic presentation, then inwx 5 there is an effective algorithm for computing TNŽ. G described. This method uses an induction upwards on the polycyclic series determined by the polycyclic presentation, and it applies cohomological methods and complement computations to lift the normal torsion subgroup through the subgroups in the polycyclic series. This normal torsion subgroup method can be extended to polycyclic-by-finite groups G. By Lemma 5.1Ž. b we can pass to GrTNŽ. G and thus assume that TN Ž. G is trivial. In this case FCŽ. G is free abelian by Lemma 5.1 Ž. a and we determine it similarly to the centre computation in Section 4 using the following theorem.

THEOREM 5.1. Let G be a polycyclic-by-finite group with TNŽ. G s 1. Then ZŽFit ŽG .. is free abelian, say Z ŽFit ŽG .. ( Z d. The natural conjugation action of G on ZŽŽ..Fit G corresponds to ␥ : G ª GL Ž d, ޚ ., and we define s ␥ и ␺ ␺ GppkerŽ.by using the p-congruence homomorphism pfor an odd prime p. Then

s FCŽ. G CGZŽFitŽG.. Ž.p .

Proof. Let U s FCŽ. G .IfTN Ž. G s 1, then U is a free abelian normal subgroup of G by Lemma 5.1 and hence U F FitŽ.G . Further, since each element in U has finite class, G acts as a finite group on U. By Theorem 2.1, FitŽ.G centralizes a series in U. Thus Fit Ž.G acts on U as an integral unitriangular group and hence as a torsion-free group. We obtain that FitŽ.G acts trivially on U, and U F Z Ž Fit Ž.. G . Let V s ZŽŽ..Fit G . Then U is the largest G-normal subgroup in V on s which G acts as a finite group. Let W CGVpŽ.. Since G phas finite index in G, we obtain that G acts on W as a finite group. Thus W F U. Note that VrU is torsion-free and thus we can choose a basis of V through U.By 184 BETTINA EICK

definition, Gp acts as a p-congruence subgroup on V and thus, since a p-congruence subgroup is invariant under base changes, Gp acts as a p-congruence subgroup on U as well. By Theorem 3.1 we obtain that Gp acts as a torsion-free group on U. But G induces a finite action on U. F Hence Gp must act trivially on U and U W. Altogether, we obtain s s U W CGVpŽ.as desired.

6. EXHIBITING THE NILPOTENT-BY-ABELIAN-BY-FINITE STRUCTURE

The following is a well-known structure theorem on polycyclic-by-finite groups which has been introduced by Mal’cev.

THEOREM 6.1. Let G be polycyclic-by-finite. Then G has a normal series

1eᎏᎏᎏM e N eG with M nilpotent, NrM abelian, and GrN finite.

This structure theorem of polycyclic-by-finite groups can be exhibited by the methods introduced above as outlined in the following theorem.

THEOREM 6.2.

Ž.a Let G be a finitely generated nilpotent group which is abelian-by- finite. Then ZŽ. G has finite index in G. Ž.b Let G be polycyclic-by-finite. Then with M s Fit ŽG . and NrM s

ZŽŽFit GrM .. we obtain a normal series 1eᎏᎏᎏM e N eG such that M is nilpotent, NrM is abelian, and GrN is finite.

Proof. Ž.a Let A ( ޚ d be a normal subgroup with finite index in G. By Theorem 2.1, G centralizes a series in A and thus acts as an integral unitriangular matrix subgroup on A. An integral unitriangular matrix group is torsion-free. Since GrA is finite, we obtain that G acts trivially on A and A is central. Ž.b Clearly, M is nilpotent and NrM is abelian. It remains to show that GrN is finite. By Theorem 6.1, H s GrM is abelian-by-finite. Thus there exists an abelian normal subgroup Aeᎏ H with finite index. Hence A F FitŽ.H and thus Fit Ž.H has finite index in H.Bya,Ž.ŽZ Fit Ž..H has finite index in FitŽ.H . Hence w H : ZŽŽ..Fit H x - ϱ, as desired.

Theorem 6.2Ž. b can readily be translated into an effective algorithm. We compute FitŽ.G as in Section 3 and pass to the factor H s GrFit Ž.G . Now it remains to determine ZŽŽ..Fit H , which is described in Lemma 4.1. FITTING SUBGROUPS 185

7. FURTHER APPLICATIONS

7.1. The Upper Nilpotent Series In the theory of finite polycyclic groups the lower nilpotent series plays a role similar to that of the lower central series of a nilpotent group. These two series have been used in the design of effective algorithms for finite polycyclic groups. However, this approach cannot be generalized to polycyclic groups in general, since the lower nilpotent series of an infinite polycyclic group might not have finite length. But the upper nilpotent series obtained by taking iterated Fitting subgroups exists in every polycyclic group, and it plays a role similar to the upper central series of a nilpotent group. It would be interesting to investigate the use of the upper nilpotent series in the algorithmic theory of infinite polycyclic groups and thus to obtain a new approach towards this theory.

7.2. An Approximation of the Frattini Subgroup The Frattini subgroup ⌽Ž.G of a polycyclic-by-finite group G is the intersection of all maximal subgroups of G. By a well-known theorem of Hirsch, ⌽Ž.G is nilpotent for a polycyclic-by-finite group G. Hence

FitŽ.G G ⌽ Ž.G G ⌽Ž.Fit Ž.G .

To determine ⌽Ž.F for a finitely generated nilpotent group F, we let FrFЈ s C = T with C free abelian and T finite abelian. Then ⌽Ž.F rFЈ s ⌽Ž.T . In particular, Fr⌽Ž.F is a abelian group. Thus we can approxi- mate ⌽Ž.G up to an abelian factor.

8. IMPLEMENTATION AND PRACTICALITY

The methods described here have been implemented in Polycyclic wx7 for polycyclic groups defined by polycyclic presentations. For background on polycyclic presentations we refer the reader towx 6 . The most time-consuming part of the outlined methods is the Fitting subgroup algorithm. This requires the computation of a generalized chief series and the determination of centralizers of each factor in such a series. The centralizer computation is usually the limiting part in the Fitting subgroup method. All further algorithms described here rely on the Fitting subgroup computation, and they are usually effective once the needed Fitting subgroup has been determined. 186 BETTINA EICK

It is difficult to give precise limits for the range of the applications of the considered methods. Clearly, the range of the application depends on the number of the orders or dimensions of the factors in the elementary or free abelian normal series chosen for the considered group. Experiments suggest that the Fitting subgroup computation is effective if the finite factors are of order at most 2 20 and the infinite factors have dimensions bounded by 8. Here we consider an application as effective if the algorithm is processing each factor of the elementary or free abelian normal series within a few seconds on a standard PC.

8.1. Applications to Polycyclic-by-Finite Groups The two most common ways to describe anŽ. infinite polycyclic-by-finite group are as a rational matrix group or by using a finite presentation. To apply our methods to a group G given in such a way, we need to determine a polycyclic presentation for its largest polycyclic normal subgroup P. If G is given as a rational matrix group, then the method of Ostheimer wx11 can be used to compute the desired information. This algorithm is expected to be practical, but an implementation is not yet available. If G is a finitely presented polycyclic group, then we can compute a polycyclic presentation for G using the method of Lowx 9, 10 or the algorithm developed in a joint work with Niemeyerwx 8 . These methods are both available in GAP. Finally, if G is polycyclic-by-finite, but not polycyclic, and G is given by a finite presentation, then a practical algorithm to determine the largest polycyclic subgroup P of G is not available so far.

8.2. Comparison to the Finite Group Case If G is a finite group, then FitŽ.G is the direct product of its Sylow subgroups. Further, the Sylow p-subgroup of FitŽ.G is the core of a Sylow p-subgroup of G in this case. This characterization is often used to determine FitŽ.G for a finite group G. Thus the Fitting subgroup method described here does not generalize this finite group method. The determination of the centre of a polycyclic-by-finite group G as outlined here is effective also in comparison with other finite group methods for this purpose. Clearly, the computation of the FC-centre or of the nilpotent-by-abelian-by-finite structure is trivial in the finite group case.

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