Elementary abelian subgroups of classical groups of type $A$

TL;DR

This paper classifies elementary abelian p-subgroups of classical groups of type A, providing insights into their structure to aid in understanding p-radical subgroups relevant to finite group representation theory.

Contribution

The paper extends the classification of elementary abelian p-subgroups from exceptional to classical groups of type A using a linear algebraic approach.

Findings

Classification of elementary abelian p-subgroups in classical groups of type A

Description of their local structure within these groups

Application of linear algebraic methods to transfer results to finite groups

Abstract

Many open conjectures in the representation theory of finite groups can be studied by reducing them to related questions about quasi-simple groups. In such studies, $p$-radical subgroups typically play a critical role. To classify the $p$-radical subgroups of a finite group, we first classify the elementary abelian $p$- subgroups and find their local structure. To do this, we conduct the classification in a linear algebraic group $G$ and then transfer the results to the finite group of Lie type $G^F$ . This approach was used by An, Dietrich and Litterick in their work on finite exceptional groups of Lie type. We now apply this approach to classical groups of type $A$ for the classification and local structure of the elementary abelian $p$-subgroups.