Order three normalizers of 2-groups

TL;DR

This paper classifies order three elements in almost simple groups that do not normalize any nontrivial 2-subgroup, providing insights relevant to modular representation theory and the structure of finite groups.

Contribution

It offers a complete classification of such elements in almost simple groups and establishes necessary conditions for their existence in broader finite groups.

Findings

Complete classification in almost simple groups

Necessary conditions for existence in finite groups

Relevance to modular representation theory

Abstract

This paper examines order three elements of finite groups which normalize no nontrivial 2-subgroup. The motivation for finding such elements arises out of a problem in modular representation theory. The question of when these elements appear in the almost simple groups was posed by Geoff Robinson in the context of studying 2-blocks of defect zero. For the almost simple groups, a complete classification of order three elements with this property is determined. On the basis of this result, necessary conditions are then given for the existence of such elements in a large class of finite groups.