Groups with finitely many conjugacy classes of subgroups with large subnormal defect

Extract

Given a group G and a positive integer k, let v_k(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that v_kG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which v_k(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v₁(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].