The Hochschild cohomology ring of a group algebra.
Stephen F. Siegel and Sarah J. Witherspoon.
Univ. of Massachusetts at Amherst
Univ. of Toronto
Preprint, submitted for publication.

Abstract:
  There is a standard additive decomposition of the Hochschild
  cohomology ring of the group algebra of a finite group $G$ as the
  direct sum of the cohomology rings of the centralizers of
  representatives of the conjugacy classes of $G$.  A special case of
  our main result describes the cup product in terms of this
  decomposition.  As applications, we determine presentations for the
  Hochschild cohomology rings of (1) the mod-$3$ group algebra of the
  symmetric group $S_3$, (2) the mod-$2$ group algebra of the
  alternating group $A_4$, and (3) the mod-$2$ group algebras of the
  dihedral $2$-groups.