Elementary abelian subgroups of classical groups of type $A$

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.