Weak chain conditions for non-almost normal subgroups

Summary

Abstract

We describe locally (soluble-by-finite) groups in which the set of all subgroups with infinitely many conjugates satisfies some weak chain condition.

AMS subject classification: 20E15, 20F16, 20F24.

Introduction

A subgroup H of a group G is said to be almost normal in G if it has finitely many conjugates in G or, equivalently, if its normalizer N_G(H) has finite index in G. A celebrated theorem by B.H. Neumann [8] states that every subgroup of a group G is almost normal if and only if the factor G/Z(G) is finite.

It is natural to ask what information on the structure of the group G can be obtained if the condition of being almost normal is imposed only on a large set of subgroups of G. For instance, I.I. Eremin [4] proved that every subgroup of G is almost normal (and so G/Z(G) is finite), provided every abelian subgroup is. Problems of this type have been considered in various papers (see [5] and references quoted therein).

A way of ensuring the existence of many almost normal subgroups is to impose some chain condition on the set of the subgroups which are not almost normal. The second author and V.V. Pylaev [7] studied groups satisfying the minimal condition on non-almost normal subgroups. Here we consider groups in which non-almost normal subgroups satisfy some weaker chain condition.