Abstract

Across group theory, elementary abelian subgroups arise naturally in many contexts. For instance, they play an important role in modular representation theory, in local structure of groups, and in the cohomology theory of various spaces.This talk will present joint work with Jianbei An (University of Auckland) and Heiko Dietrich (Monash University, Melbourne), in which we consider elementary abelian subgroups of reductive algebraic groups in positive characteristic. In contrast with previous works which proceed ‘bottom up’, beginning with elements of order p, then elements of order p in their centralisers, and so on, we use a ‘top-down’ approach building on work of R. Griess on maximal elementary abelian subgroups and their normaliser structure. Such subgroups behave differently depending on whether or not they are toral (contained in a torus), and our results are two-fold. For toral subgroups, we give an efficient combinatorial algorithm for enumerating subgroups and determining their normaliser and centraliser structure. For non-toral subgroups, we complement work of J. Yu and Andersen et. al., and end up with a complete classification of subgroups which is independent of the ambient characteristic. The eventual aim is to use these results to prove local structure results in finite groups of Lie type, via the Lang-Steinberg theorem; I will close with a discussion of the subtleties arising in this process.