Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture

TL;DR

This paper investigates the structure of locally finite groups with bounded chains of centralizers, disproving a conjecture that such groups' quotients contain large abelian subgroups, and explores related weaker results.

Contribution

The paper disproves the Borovik--Khukhro conjecture and establishes some weaker analogs regarding the structure of groups with bounded centralizer chains.

Findings

Disproved the Borovik--Khukhro conjecture.

Established weaker analogs for groups with bounded centralizer chains.

Provided structural insights into locally finite groups with bounded centralizer chains.

Abstract

Let $G$ be a locally finite group and $F(G)$ the Hirsch--Plotkin radical of $G$. Denote by $S$ the full inverse image of the generalized Fitting subgroup of $G/F(G)$ in $G$. Assume that there is a number $k$ such that the length of every chain of nested centralizers in $G$ does not exceed $k$. The Borovik--Khukhro conjecture states, in particular, that under this assumption the quotient $G/S$ contains an abelian subgroup of index bounded in terms of $k$. We disprove this statement and prove some its weaker analog.