Title: The essential ideal in group cohomology does not square to zero
Author: David J. Green 
MSC: 20J06 (primary) 57S17 (secondary)
arXiv: math.GR/0302336
Status: Submitted for publication, February 2003

Abstract:
Let G be the Sylow 2-subgroup of the unitary group $SU_3(4)$. We find two
essential classes in the mod-2 cohomology ring of G whose product is nonzero.
In fact, the product is the ``last survivor'' of Benson--Carlson duality.
Recent work of Pakianathan and Yalcin then implies a result about connected
graphs with an action of G. Also, there exist essential classes which cannot be
written as sums of transfers from proper subgroups.
  This phenomenon was first observed on the computer. The argument given here
uses the elegant calculation by J. Clark, with minor corrections.