Maximal cocliques and cohomology in rank one linear groups

Abstract

In this thesis, we investigate certain aspects of $\mathrm{PSL}_2(q)$. We begin by looking at the generating graph of $\mathrm{PSL}_2(q)$, a structure which may be used to encode certain information about the group, which was first introduced by Liebeck and Shalev and further investigated by many others. We provide a classification of maximal cocliques (independent sets) in the generating graph of $\mathrm{PSL}_2(q)$ when $q$ is a prime and provide a family of examples to show that this result does not directly extend to the prime-power case. After this, we instead investigate the cohomology of finite groups and prove a general result relating the first cohomology of any module to the structure of some fixed module and a generalisation of this result to higher cohomology. We then completely determine the cohomology $\mathrm{H}^n(G,V)$ and its generalisation, $\mathrm{Ext}_G^n(V,W)$, for irreducible modules $V$, $W$ for $G = \mathrm{PSL}_2(q)$ for all $q$ in all non-defining characteristics before doing the same for the Suzuki groups