Title: Primitives and central detection numbers in group cohomology
Author: Nicholas J. Kuhn
Address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903
email: njk4x@virginia.edu
AMS classification numbers: 20J06, 55R40
ArXiv number: math.GR/0612133
abstract:
Henn, Lannes, and Schwartz have introduced two invariants, d_0(G) and
d_1(G), of the mod p cohomology of a finite group G such that H^*(G) is
detected and determined by H^d(C_G(V)) for d no bigger than d_0(G) and
d_1(G), with V < G p-elementary abelian. We study how to calculate these
invariants.
We define a number e(G) that measures the image of the restriction of
H^*(G) to its maximal central p-elementary abelian subgroup. When G has
a p-central Sylow subgroup P, d_0(G) = d_0(P) = e(P), and a similar
exact formula holds for d_1(G). In general, we show that d_0(G) is
bounded above by the maximum of the e(C_G(V))'s, if Benson's Regularity
Conjecture holds. In particular, this holds for all groups such that
the p--rank of G minus the depth of H^*(G) is at most 2. When we look
at examples with p=2, we learn that d_0(G) is at most 7 for all groups
with 2--Sylow subgroup of order up to 64, unless the Sylow subgroup is
isomorphic to that of either Sz(8) (and d_0(G) = 9) or SU(3,4) (and
d_0(G)=14).
Enroute we recover and strengthen theorems of Adem and Karagueuzian on
essential cohomology, and Green on depth essential cohomology, and prove
theorems about the structure of cohomology primitives associated to
central extensions.