Locally finite groups of finite $c$-dimension

TL;DR

This paper investigates the structure of locally finite groups with finite c-dimension, establishing key theorems and bounds related to their centralizer chains and quotients.

Contribution

It provides two structure theorems for these groups and bounds the c-dimension of their quotients by locally soluble radicals.

Findings

Proved structure theorems for locally finite groups with finite c-dimension.

Established bounds on c-dimension of quotients by locally soluble radicals.

Connected c-dimension properties to group structure and quotients.

Abstract

The supremum of lengths of strict chains of nested centralizers is called the $c$-dimension (centralizer dimension) of $G$. We prove two structure theorems for locally finite groups of finite $c$-dimension. We also prove that the $c$-dimension of the quotient of such a group by a locally soluble radical is bounded in terms of the $c$-dimension of the group itself.