Groups whose proper subgroups are locally ﬁnite-by-nilpotent

If X is a class of groups, then a group G is said to be minimal non X -group if all its proper subgroups are in the class X , but G itself is not an X -group. The main result of this note is that if c > 0 is an integer and if G is a minimal non ( LF ) N (respectively, ( LF ) N c )-group, then G is a ﬁnitely generated perfect group which has no non-trivial ﬁnite factor and such that G/Frat ( G ) is an inﬁnite simple group; where N (respectively, N c , LF ) denotes the class of nilpotent (respectively, nilpotent of class at most c , locally ﬁnite) groups and Frat ( G ) stands for the Frattini subgroup of G


Introduction
If X is a class of groups, then a group G is said to be minimal non-X if all its proper subgroups are in the class X , but G itself is not an Xgroup.Many results have been obtained by many authors on minimal non X -groups for various classes of groups X , for example see [1], [2], [3], [5], [6], [8], [11], [12], [13].In particular, in [13] it is proved that if G is a finitely generated minimal non FN -group, then G is a perfect group which has no non-trivial finite factor and such that G/F rat(G) is an infinite simple group; where N (respectively, F) denotes the class of nilpotent (respectively, finite) groups and F rat(G) stands for the Frattini subgroup of G.The aim of the present note is to extend the above results to minimal non (LF)N (respectively, (LF)N c )-groups, and to prove that there are no minimal non (LF)N (respectively, (LF)N c )-groups which are not finitely generated; where c > 0 is an integer and N c (respectively, LF) denotes the class of nilpotent groups of class at most c (respectively, locally finite groups).More precisely we shall prove the following results.
Theorem 1.1.If G is a minimal non (LF)N -group, then G is a finitely generated perfect group which has no non-trivial finite factor and such that G/F rat(G) is an infinite simple group.
Using Theorem 1.1, we shall prove the following result on minimal non (LF)N c -groups.Theorem 1.2.Let c > 0 be an integer and let G be a minimal non (LF)N c -group.Then G is a finitely generated perfect group which has no non-trivial finite factor and such that G/F rat(G) is an infinite simple group.
Note that if X 1 and X 2 are two classes of groups such that X 1 ⊆ X 2 , then a minimal non X 1 -group is either a minimal non X 2 -group or an X 2 -group.From Xu's results [13,Theorem 3.5], an infinitely generated minimal non FN -group is a locally finite-by-nilpotent group.So one might expect, as we shall prove in Proposition 2.1, that there are no infinitely generated minimal non (LF)N -group.
Note that minimal non (LF)N (respectively, non (LF)N c )-groups exist.Indeed, the group constructed by Ol'shanskii [7] is a simple torsion-free finitely generated group whose proper subgroups are cyclic.

Minimal non (LF)N -groups
A part of Theorem 1.1 is an immediate consequence of the following Proposition: Proposition 2.1.Let G be a group whose proper subgroups are in the class (LF)N .Then G belongs to (LF)N if it satisfies one of the following two conditions: (i) G is finitely generated and has a proper subgroup of finite index, (ii) G is not finitely generated.
Proof.(i) Suppose that G is finitely generated and let N be a proper subgroup of finite index in G.By [10, Theorem 1.6.9]we may assume that N is normal in G.So N is in (LF)N and it is also finitely generated.Hence γ k+1 (N ) is locally finite for some integer k ≥ 0. Since N is of finite index in G, G/γ k+1 (N ) is a finitely generated group in the class N F, so that it satisfies the maximal condition on subgroups.Therefore every proper subgroups of G/γ k+1 (N ) is in FN .Now Lemma 4 of [3] states that a finitely generated locally graded group whose proper subgroups are finiteby-nilpotent is itself finite-by-nilpotent.Since groups in the class N F are clearly locally graded, we deduce that (ii) Suppose that G is not finitely generated and let x 1 , ..., x n be n elements of finite order in G. Since the subgroup x 1 , ..., x n is proper in G, it is in (LF)N , hence it is finite.This means that the elements of finite order in G form a locally finite subgroup T .If G/T is not finitely generated, then it is locally nilpotent and its proper subgroups are nilpotent as G/T is torsion-free.Now Theorem 2.1 of [11] states that a torsion-free locally nilpotent group with all proper subgroups nilpotent is itself nilpotent.Therefore G/T is nilpotent, so G is in (LF)N .Now if G/T is finitely generated, then there exists a finitely generated subgroup H such that Since finitely generated locally graded groups have proper subgroups of finite index, the previous Proposition admits the following consequence : Proof of Theorem 1.1.Let G be a minimal non (LF)N -group.It follows from Proposition 2.1 and Corollary 2.3 that G is a finitely generated perfect group which has no non trivial finite factor.Now we prove that G/F rat(G) is an infinite simple group.Since finitely generated groups have maximal subgroups, G/F rat(G) is non trivial and therefore infinite.Let N be a proper normal subgroup of G properly containing F rat(G).Then N is in (LF)N and there is an x ∈ N such that x / ∈ F rat(G).Hence there is a maximal subgroup M of G such that x / ∈ M , so N is not contained in M .Then G = N M and we have γ k+1 (M ) is locally finite for some integer k ≥ 0. Since G is perfect, then We show by induction on k that γ k+1 (N M ) ⊆ N γ k+1 (M ).If k = 0, then the result follows immediately.Now let k > 0, suppose inductively that γ k (N M ) ⊆ N γ k (M ) and let g be an element of γ k+1 (N M ).Hence g can be written as a finite product of elements of the form [x 1 y 1 , ..., x k+1 y k+1 ] with x i ∈ N and y i ∈ M for every 1 ≤ i ≤ k + 1.It follows by the inductive hypothesis that the commutator v = [x 1 y 1 , ..., x k y k ] of weight k is in N γ k (M ).So we get [v, x k+1 y k+1 ] = [xy, x k+1 y k+1 ] with x ∈ N and y ∈ γ k (M ).Therefore We have that [y, y k+1 ] is in γ k+1 (M ) and since N is normal in G, we have that [x, y k+1 ] y and ([x, x k+1 ] y [y, x k+1 ]) y k+1 belong to N .Thus [x 1 y 1 , ..., x k+1 y k+1 ] is in N γ k+1 (M ) and consequently g belongs to N γ k+1 (M ).

Hence the inclusion γ
, where A denotes the class of abelian groups; so G/N is a locally graded group.We deduce from Corollary 2.2 that G/N is in (LF)N .Now Theorem 1.2 of [4] states that if N G such that N and G/N are in (LF)N , then G is in (LF)N .So G is in (LF)N , a contradiction.This means that G/F rat(G) is a simple group.Proof of Theorem 1.2.Let G be a minimal non (LF)N c -group.It follows that every proper subgroup of G is in (LF)N .Now suppose that G is in (LF)N , so there exists a normal subgroup N of G such that N is locally finite and G/N is nilpotent.By Corollary 3.3, G/N is in (LF)N c , consequently we deduce that G is in (LF)N c because N is locally finite; a contradiction.Hence G is a minimal non (LF)N -group, and by Theorem 1.1, G is a finitely generated perfect group which has no non-trivial finite factor and such that G/F rat(G) is an infinite simple group.

Corollary 2 . 2 .Corollary 2 . 3 .
Let G be a locally graded group whose proper subgroups are in the class (LF)N .Then G is in the class (LF)N .Let G be a non perfect group whose proper subgroups are in the class (LF)N .Then G is in the class (LF)N .Proof.If G is not finitely generated, then G is in (LF)N from (ii) of Proposition 2.1.Now suppose that G is finitely generated.Therefore G/G , A. Dilmi being a non trivial finitely generated locally graded group, has a non trivial finite image.So G has a proper subgroup of finite index.Thus we deduce from (i) of Proposition 2.1 that G is in (LF)N .

Corollary 3 . 3 .
Since finitely generated locally graded have proper subgroups of finite index, Proposition 3.2 admits the following consequence : Let c > 0 be an integer and let G be a locally graded group whose proper subgroups are in the class (LF)N c .Then G is in the class (LF)N c .