THE NILPOTENT FILTRATION AND THE ACTION OF AUTOMORPHISMS ON THE COHOMOLOGY OF FINITE p–GROUPS

We study H∗(P ), the mod p cohomology of a finite p–group P , viewed as an Fp[Out(P )]–module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if eS ∈ Fp[Out(P )] is a primitive idempotent associated to an irreducible Fp[Out(P )]–module S, then the Krull dimension of eSH ∗(P ) equals the rank of P . The rank is an upper bound by Quillen’s work, and the conjecture can be viewed as the statement that every irreducible Fp[Out(P )]–module occurs as a composition factor in H∗(P ) with similar frequency. In summary, our results are as follows. A strong form of the conjecture is true when p is odd. The situation is much more complex when p = 2, but is reduced to a question about 2–central groups (groups in which all elements of order 2 are central), making it easy to verify the conjecture for many finite 2–groups, including all groups of order 128, and all groups that can be written as the product of groups of order 64 or less. The the odd prime theorem can be deduced using the approach to U , the category of unstable modules over the Steenrod algebra, initiated by H.-W. Henn, J. Lannes, and L. Schwartz in [HLS1]. The reductions when p = 2 make heavy use of the nilpotent filtration of U introduced in [S1], as applied to group cohomology in [HLS2]. Also featured are unstable algebras of cohomology primitives associated to central group extensions.