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

Abstract

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

arXiv comment30 pages, updated to published version

References in corpus (1)

• First cohomology groups for finite groups of Lie type in defining characteristic