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

Abstract

Abstract. 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 e S ∈ Fp[Out(P )] is a primitive idempotent associated to an irreducible Fp[Out(P )]-module S, then the Krull dimension of e S H * (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.

Primitives and central detection numbers in group cohomology

“…In the p-central case, we also input Carlson and Benson's theorem that if H * (G) is Cohen-Macauley then it is Gorenstein: this leads to proofs of Theorems 2.8 and 2.9 in Section 7. Using an analysis of the formula in Theorem 2.5 done by us in [28], Theorem 2.14 is proved in Section 8, which then continues with our results about Cess * (G) and the conjectured inequality e (G) e(G). Though short examples occur throughout, some longer examples that illustrate the general theory make up Section 9.…”

“…To understand d 0 (G) for general G, one needs to use the more complicated formula given in Theorem 2.5. Using some analysis of this already done by us in our companion paper [28], we are led to a formula 6 for d 0 (G) that makes use of the following variant of essential cohomology.…”

Equivariant Hilbert series

“…In this section we apply some of the theory that we have developed to a problem in group cohomology considered by Nick Kuhn [2008]. We fix a prime p and a finite group P (we do not yet require P to be a p-group).…”

“…Given a p-group P and a simple G-module V , Martino and Priddy [1992] asked whether the dimension of V as a composition factor of H * (P) is equal to dim H * (P) (see also [Kuhn 2008]). It was already known from [Diethelm and Stammbach 1984;Harris and Kuhn 1988;Symonds 1999] that V does occur in…”

“…Theorem 8.1 [Kuhn 2008]. For p odd, the dimension of V as a composition factor of H * (P) is equal to the dimension of H * (P).…”