The Structure of SL₂(q) and its Modules

Abstract

Partly because in the definition of a symplectic amalgam (see Definition 1.2) we have L ₁/O _p (L ₁) ≅ SL₂(q), the group SL ₂(q) and its GF(p)-modules play a pivotal role in the exploration of these amalgams. For this chapter we fix q = p ^a, k = GF(q) and X = SL₂(q). The chapter is divided into two sections. The first section deals with the subgroup structure and covering groups of X. These results are for the most part well-known and good sources are [61] and [79]. The second section is devoted to kX-modules and we begin by presenting a description of all irreducible kX-modules as given by Brauer and Nesbitt in [22]. Then, using results on generation of X by p-elements, we characterize those GF(p)X-modules that have small order. We further determine the GF(p)X-modules on which a Sylowp-subgroup ofX can operate quadratically. The chapter closes with a module result for SL₂(2) ≀ 2