Profinite groups with soluble centralisers

TL;DR

This paper characterizes profinite groups with specific centralizer properties, showing they are either virtually pro-p or virtually soluble with bounded derived length, advancing understanding of their structure.

Contribution

It establishes new structural classifications for profinite groups based on the properties of their element centralizers.

Findings

Profinite groups with metabelian centralizers are either virtually pro-p or virtually soluble of derived length ≤ 4.

Prosoluble groups with soluble centralizers of bounded derived length are either soluble or virtually pro-p.

Abstract

We show that a profinite group, in which the centralisers of non-trivial elements are metabelian, is either virtually pro-$p$ or virtually soluble of derived length at most 4. We furthermore show that a prosoluble group, in which the centralisers of non-trivial elements are soluble of bounded derived length, is either soluble or virtually pro-$p$.