Cohomology of $\text{PSL}_2(q)$

Jack Saunders

Published 2020-02-11Version 1

In 2011, Guralnick and Tiep proved that if $G$ was a Chevalley group and $V$ an irreducible $G$-module in cross characteristic, then if $V^B = 0$, the dimension of $H^1(G,V)$ is determined by the structure of the permutation module on a Borel subgroup $B$ of $G$. We generalise this theorem to higher cohomology and an arbitrary finite group, so that if $H \leq G$ such that $O_{r'}(H) = O^r(H)$ then if $V^H = 0$ we show $\dim H^1(G,V)$ is determined by the structure of the permutation module on $H$, and $H^n(G,V)$ by $\text{Ext}_G^{n-1}(V^*,M)$ for some $M$ dependent on $H$. We also explicitly determine $\text{Ext}_G^n(V,W)$ for all irreducible $kG$-modules $V$, $W$ for $G = \text{PSL}_2(q)$ in cross characteristic.