On finite groups with soluble centralisers

TL;DR

This paper classifies finite groups based on the solubility of centralisers of certain elements, providing structural insights and reduction theorems with applications to groups with soluble involution centralisers and non-commuting graphs.

Contribution

It offers a comprehensive classification of finite groups with soluble centralisers of specific elements and introduces reduction theorems for groups with all non-central $ ext{pi}$-element centralisers.

Findings

Full structural description of groups with all non-central element centralisers soluble

Reduction theorem for groups where all non-central $ ext{pi}$-element centralisers are soluble

Applications to groups with soluble involution centralisers and non-commuting graphs

Abstract

We classify finite groups in which the centralisers of certain non-central elements are soluble. This includes a full structural description of groups whose non-central element centralisers are all soluble, and a reduction theorem for the case in which all non-central $\pi$-elements have soluble centralisers, for a suitable collection $\pi$ of primes. Our results yield further descriptions under mild local conditions and have applications to groups with soluble involution centralisers, as well as to questions concerning non-commuting graphs.