The Nilpotent filtration and the action of automorphisms on the cohomology of finite p-groups

Abstract We study H*(P), the mod p cohomology of a finite p-group P, viewed as an $\F_p[Out(P)]$–module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e ∈ $\F_p[Out(P)]$ is a nonzero idempotent, then the Krull dimension of eH*(P) equals the rank of P. We prove this for all p-groups when p is odd, and for many 2–groups.