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 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 GL_n F_(p^r) , when p = 2 or r = 1, we calculate the cohomology in degree r(2p − 3), for all n.