2006/05/16 by Nicholas J. Kuhn, Kuhn, Nicholas J.
Mathematics · #20D15 #20J06 #55R40 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.AT #math.GR #msc:20D15 #msc:20J06 #msc:55R40
paper · pdf · doi:10.48550/arxiv.math/0605442
29 pages
arxiv created 2006/05/16 · openalex publication_date 2006/05/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study H^*(P), the mod p cohomology of a finite p--group P, viewed as an Out(P)--module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if eS in Z/p[Out(P)] is a primitive idempotent associated to an irreducible Z/p[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 Z/p[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. Featured is the nilpotent filtration of the category of unstable A--modules. Also featured are unstable algebras of cohomology primitives associated to central group extensions.