Primitives and central detection numbers in group cohomology |
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Authors: | Nicholas J Kuhn |
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Institution: | Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA |
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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 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<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).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. |
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Keywords: | primary 20J06 secondary 55R40 |
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