  Staic defined symmetric cohomology of groups and studied that the secondary
symmetric cohomology group is corresponding to group extensions and the
injectivity of the canonical map from symmetric cohomology to classical
cohomology. In this paper, we define symmetric cohomology and symmetric
Hochschild cohomology for cocommutative Hopf algebras. The first one is a
generalization of symmetric cohomology of groups. We give an isomorphism
between symmetric cohomology and symmetric Hochschild cohomology, which is a
symmetric version of the classical result about cohomology of groups by
Eilenberg-MacLane and cohomology of Hopf algebras by Ginzburg-Kumar. Moreover,
to consider the condition that symmetric cohomology coincides with classical
cohomology, we investigate the projectivity of a resolution which gives
symmetric cohomology.
