Title: Primitives and central detection numbers in group cohomology
Author: Nicholas J. Kuhn
Address: Department of Mathematics, University of Virginia, 
Charlottesville, VA 22903
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 somorphic 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.