On finite groups with soluble centralisers
Valentina Grazian, Carmine Monetta, Gareth Tracey
TL;DR
The paper addresses the problem of classifying finite groups whose non-central centralisers are soluble, with a primary focus on the case where all such centralisers are soluble and on reduction to the $\pi$-element setting. It develops a cohesive framework combining cohomology (vanishing results for $H^1$ and $H^2$), Frattini lifting, and embeddings of alternating groups into Lie-type and sporadic groups, along with computational verification. The main contributions include a complete description of $G/R(G)$ when all non-central centralisers are soluble, and a reduction theorem for the $\pi$-element case that constrains non-abelian composition factors to specific families and bounds their complexity (twisted rank in Lie types and finite alternating bounds). The results yield applications to groups with soluble involution centralisers and to non-commuting graphs, including a graph-theoretic reduction for minimal counterexamples and explicit thickness bounds for alternating sections. Overall, the work provides a unified, technically diverse approach that clarifies how local centraliser solubility imposes strong global restrictions on a group’s structure.
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 $π$-elements have soluble centralisers, for a suitable collection $π$ 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.