  We construct a Frobenius algebra structure on the Hochschild cochains of a
group ring k[G] that extends the known structure of a <1, 2> topological
quantum field theory on HH^0(k[G]; k[G]), k a field and G a finite group. The
convolution product extends to the homotopy commutative Gerstenhaber product on
cochains, the Frobenius coproduct extends to a coproduct on the chain complex
for Hochschild homology, and there is a pairing on Hochschild cocahins
satisfying Frobenius associativity. The pairing, however, degenerates on a
certain subcomplex of Hochschild cochains. The cochain complex for group
cohomology under the simplicial cup product occurs as a homotopy commutative
subalgebra of the Hochschild cochain complex under the Gerstenhaber product.
