Cohomology of PSL2(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 VB=0, then the dimension of H1(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≤G such that Or′(H)=Or(H) and VH=0 for V a G-module in characteristic r then dim⁡H1(G,V) is determined by the structure of the permutation module on cosets of H, and Hn(G,V) by ExtGn−1(V⁎,M) for some kG-module M dependent on H. We also determine ExtGn(V,W) for all irreducible kG-modules V, W for G∈{PSL2(q),PGL2(q),SL2(q)} in cross characteristic.