A finiteness condition on centralizers in locally finite groups

TL;DR

This paper characterizes locally finite groups where each non-normal cyclic subgroup has a finite or bounded index in its centralizer, revealing they are cyclic extensions of Dedekind groups and extending to periodic locally graded groups.

Contribution

It provides a complete description of groups satisfying a finiteness condition on centralizers, including a variation with bounded index, expanding understanding of group structure under these conditions.

Findings

Locally finite groups with finite centralizer indices are cyclic extensions of Dedekind groups.

The analysis extends to periodic locally graded groups with bounded centralizer index.

The paper classifies groups satisfying the finiteness and boundedness conditions on centralizers.

Abstract

We consider a finiteness condition on centralizers in a group G, namely that |C_G (x) : <x>| is finite for every non-normal cyclic subgroup <x> of G. For periodic groups, this is the same as |C_G (x)| is finite for every non-normal cyclic subgroup <x> of G. We give a full description of locally finite groups satisfying this condition. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study a variation of our condition, where the requirement of finiteness is replaced with a bound: |C_G (x) : <x>| < n for every non-normal cyclic subgroup <x> of G, for some fixed n. In this case, we are able to extend our analysis to the class of periodic locally graded groups.