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 1990s, 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 Hd(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 the 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