Subnormalisers of semisimple elements in finite groups of Lie type

TL;DR

This paper characterizes subnormalisers of prime power order semisimple elements in finite Lie type groups, advancing understanding of their subgroup structure with implications for character theory and permutation group properties.

Contribution

It provides a detailed determination of subnormalisers of semisimple elements and maximal overgroups of Sylow tori in finite groups of Lie type, addressing open conjectures.

Findings

Identified subnormalisers of semisimple elements of prime power order.

Determined maximal overgroups of Sylow tori.

Contributed to the understanding of subgroup structure related to character correspondence.

Abstract

We determine subnormalisers of semisimple elements of prime power order in finite quasi-simple groups of Lie type. For this, we determine the maximal overgroups of normalisers of Sylow tori. This is motivated by the recent character correspondence conjecture by Moret\'o and Rizo as well as by the question of existence of quasi-semiregular elements in finite permutation groups.