Primitives and Central Detection Numbers in Group Cohomology

Fix a prime p. Given a finite group G, let H(G) denote its mod p cohomology. In the early 1990’s, Henn, Lannes, and Schwartz introduced two invariants d0(G) and d1(G) of H∗(G) viewed as a module over the mod p– Steenrod algebra. They showed that, in a precise sense, H∗(G) is respectively detected and determined by H(CG(V )) for d ≤ d0(G) and d ≤ d1(G), with V running through the elementary abelian p–subgroups of G. The main goal of this paper is to study how to calculate these invariants. We find that a critical role is played by the image of the restriction of H∗(G) to H∗(C), where C is the maximal central elementary abelian p–subgroup of G. A measure of this is the top degree e(G) of the finite dimensional Hopf algebra H∗(C)⊗H∗(G) Fp, a number that tends to be quite easy to calculate. Our results are complete when G has a p–Sylow subgroup P in which every element of order p is central. Using Benson–Carlson duality, we show that in this case, d0(G) = d0(P ) = e(P ), and a similar exact formula holds for d1. As a bonus, we learn that He(G)(P ) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian. In general, we are able to show that d0(G) ≤ max{e(CG(V )) | V < G} if certain cases of Benson’s Regularity Conjecture hold. In particular, this inequality holds for all groups such that the difference between the p–rank of G and the depth of H∗(G) is at most 2. When we look at examples with p = 2, we learn that d0(G) ≤ 14 for all groups with 2–Sylow subgroup of order up to 64, with equality realized when G = SU(3, 4). Enroute we study two objects of independent interest. If C is any central elementary abelian p–subgroup of G, then H∗(G) is a H∗(C)–comodule, and we prove that the subalgebra of H∗(C)–primitives is always Noetherian of Krull dimension equal to the p–rank of G minus the p–rank of C. If the depth of H∗(G) equals the rank of Z(G), we show that the depth essential cohomology of G is nonzero (reproving and extending a theorem of Green), and Cohen–Macauley in a certain sense, and prove related structural results.