Some cohomology of finite general linear groups

Abstract

We prove that the degree $r(2p-3)$ cohomology of any (untwisted) finite group of Lie type over ${\mathbb{F}}_{p^r}$, with coefficients in characteristic $p$, is nonzero as long as its Coxeter number is at most $p$. We do this by providing a simple explicit construction of a nonzero element. Furthermore, for the groups $G{L}_n{\mathbb{F}}_{p^r}$, when $p=2$ or $r=1$, we calculate the cohomology in degree $r(2p-3)$, for all $n$.

Dissertation, at the University of Washington