  Symmetric homology is an analog of cyclic homology in which the cyclic groups
are replaced by symmetric groups. The foundations for the theory of symmetric
homology of algebras are developed in the context of crossed simplicial groups
using derived functors and the symmetric bar construction of Fiedorowicz. The
symmetric homology of group rings is related to stable homotopy theory. Two
chain complexes are constructed that compute symmetric homology, as well as two
spectral sequences. In the setup of the second spectral sequence, a complex
isomorphic to the suspension of the cycle-free chessboard complex of Vrecica
and Zivaljevic appears. Homology operations are defined on the symmetric
homology groups over Z/p, p a prime. Finally, an explicit partial resolution is
presented, permitting the computation of the zeroth and first symmetric
homology groups of finite-dimensional algebras.
