Primitives and central detection numbers in group cohomology

Abstract

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 d 0 ( G ) and d 1 ( 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 d ( C G ( V ) ) for d ⩽ d 0 ( G ) and d ⩽ d 1 ( 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 ) F p , 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, d 0 ( G ) = d 0 ( P ) = e ( P ) , and a similar exact formula holds for d 1 . As a bonus, we learn that H e ( G ) ( P ) contains nontrivial essential cohomology, reproving and sharpening a theorem of Adem and Karagueuzian. In general, we are able to show that d 0 ( G ) ⩽ max { e ( C G ( 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 d 0 ( G ) ⩽ 14 for all groups with 2-Sylow subgroup of order up to 64, with equality realized when G = SU ( 3 , 4 ) . En route we study two objects of independent interest. If C is any central elementary abelian p-subgroup of G, then H ∗ ( G ) is an 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.