A remark on the definability of the Fitting subgroup and the soluble radicalby Abderezak Ould Houcine

Mathematical Logic Quarterly

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Year
2013
DOI
10.1002/malq.201200038
Subject
Logic

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Math. Log. Quart. 59, No. 1–2, 62 – 65 (2013) / DOI 10.1002/malq.201200038

A remark on the definability of the Fitting subgroup and the soluble radical

Abderezak Ould Houcine∗

Mathematisches Institut und Institut fu¨r Mathematische Logik und Grundlagenforschung, Fachbereich

Mathematik und Informatik, Universita¨t Mu¨nster, Einsteinstrasse 62, 48149 Mu¨nster, Germany ´Ecole Centrale de Lyon, 6 avenue Guy de Collongue, 69134 Ecully cedex, France

Institut Camille Jordan, Universite´ Claude Bernard Lyon 1, 43 blvd du 11 novembre 1918, 69622 Villeurbanne cedex, France

Received 2 May 2012, accepted 13 August 2012

Published online 30 January 2013

Key words Group, definable, Fitting subgroup, soluble radical.

MSC (2010) 20F14, 20F16, 20F17, 20F18, 20F19, 03Cxx

Let G be an arbitrary group. We show that if the Fitting subgroup of G is nilpotent then it is definable. We prove also that the class of groups whose Fitting subgroup is nilpotent of class at most n is elementary. We give an example of a group (arbitrary saturated) whose Fitting subgroup is definable but not nilpotent. Similar results for the soluble radical are given; that is for the subgroup generated by all normal soluble subgroups of G. c© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction

Let G be a group. The Fitting subgroup of G is the subgroup generated by all normal nilpotent subgroups of G. It will be denoted F (G). The subgroup generated by all normal soluble subgroups of G is called the soluble radical and will be denoted R(G).

We see that if G is finite then F (G) is nilpotent and R(G) is soluble but in general F (G) is not necessarily nilpotent, similarly R(G) is not necessarily soluble. However, we can see that for an arbitrary group G, F (G) is locally nilpotent and R(G) is locally soluble.

It is well-known that the Fitting subgroup (or the soluble radical) is definable in many classes of groups; for instance in the class of groups of finite Morley rank or more generally in the class of superstable groups. However it is not the case in other groups and for instance the problem of the definability of the soluble radical in stable groups is steal open.

The definability of the Fitting subroup (or the soluble radical) of a group G makes it more accessible to use model-theoretic techniques. The purpose of the present note is to give relations between the nilpotency and the definability of the Fitting subgroup (we treat also the soluble radical). The first observation is that the nilpotency (resp. the solubility) of the Fitting subgroup (resp. of the soluble radical) implies its definability.

Theorem 1.1 (1) For any n ≥ 1, there exists a formula φn (x), depending only on n, such that for any group G, if F (G) is nilpotent of class n then it is definable by φn . (2) For any n ≥ 1, there exists a formula ψn (x), depending only on n, such that for any group G, if R(G) is of derived length n then it is definable by ψn .

One may ask if the given formula can define the Fitting subgroup in elementary extensions of G. We show that it is the case. ∗ e-mail: ould@math.univ-lyon1.fr c© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Math. Log. Quart. 59, No. 1–2 (2013) / www.mlq-journal.org 63

Theorem 1.2 The class of groups whose Fitting subgroup is nilpotent of class at most n is elementary. Similarly for the class of groups whose soluble radical is of derived length at most n.

One of the consequences of Theorems 1.1 and 1.2 is that if G is anMC -group then for any group K elementary equivalent to G, F (K) is nilpotent and definable. Indeed by a result of Derakhshan and Wagner [3, 6] F (G) is nilpotent and we apply Theorems 1.1 and 1.2. We note that being anMC -group is not necessarily conserved by elementary equivalence. Recently Altı´nel and Baginski [1] showed that in anMC -group every nilpotent subgroup is contained in a definable nilpotent subgroup.

As noticed above F (G) is locally nilpotent. In fact F (G) satisfies a stronger property: the normal closure (in

G) of any finite subset of F (G) is nilpotent. To formulate the next result we introduce the following definition: we say that F (G) is uniformly normaly locally nilpotent if for any m there exists d(m) such that for any finite subset A ⊆ F (G) of cardinality at most m the normal closure of A is nilpotent of class at most d(m). The notion of R(G) is uniformly normaly locally soluble is defined in a similar way.

Theorem 1.3 Let G be an ℵ0-saturated group. Then F (G) is definable if and only if F (G) is uniformly normaly locally nilpotent. Similarly R(G) is definable if and only if R(G) is uniformly normaly locally soluble.

One may also ask if the converse of Theorem 1.1 is true, that is if the definability of the Fitting subgroup implies its nilpotency (similarly for the soluble radical). At the end of the next section we give an example, based on a result of Bachmuth and Mochizuki [2], of a group G (even ℵ0-saturated) such that F (G) is definable but not nilpotent, similarly for the soluble radical. 2 Proofs

The notations which we are going to use are very standard. Given a group G and g, h ∈ G the commutator [g, h] is defined to be g−1h−1gh. We use the notation gh = h−1gh. If A is a subset of G, then 〈A〉 denotes the subgroup generated by A and 〈A〉G stands for the normal closure of A. For g ∈ G, gG denotes the conjugacy class of g.

We see that 〈gG 〉 = 〈g〉G . If H,K are subgroups of G then [H,K] is the subgroup generated by the commutators [h, k] where h ∈ H, k ∈ K.

The next lemma can be deduced from [5] but for the sake of completeness we provide a proof.

Lemma 2.1 Let G be a group, H and K be normal subgroups of G. Suppose that H is generated by A and

K is generated by B. Then [H,K] is generated by X = {[ aα , bβ ] : a ∈ A, b ∈ B,α, β ∈ G}.

P r o o f. We note that 〈X〉 is normal and 〈X〉 ≤ [H,K]. We must show that for any h ∈ H, k ∈ K, [h, k] ∈ 〈X〉. There exists a word w (resp. v) over the alphabet A±1 (resp. over B±1) such that h = w(a¯), k = v(b¯). The result follows by induction on |w|+ |v|, where |.| denotes the word-length, by using the following formulae: [x, yz] = [x, z][x, y]z , [xy, z] = [x, z]y [y, z], [x, y]z = [xz , yz ],[ x−1 , y ] = [ x, yx −1 ]−1 , [ x, y−1 ] = [ xy −1 , y ]−1 .