Group cohomology of symmetric groups

Classifying space and corresponding chain complex
The homology and cohomology groups of the symmetric group $$S_n$$ are the same as the respective homology and cohomology groups of the configuration space of $$n$$ unordered points in a countable-dimensional real projective space. For more on the topological perspective, see configuration space of unordered points of a countable-dimensional real projective space on the Topospaces wiki.

Over the integers
Here, "0" for a group is shorthand for the trivial group. $$\mathbb{Z}_m$$ is shorthand for the finite cyclic group $$\mathbb{Z}/m\mathbb{Z}$$.

The homology groups eventually stabilize in the following sense: for any fixed $$q$$, there exists a large enough $$n$$ (explicit expression for $$n$$ in terms of $$q$$ -- around double?) such that $$H_q(S_m;\mathbb{Z}) \cong H_q(S_n;\mathbb{Z})$$ for all $$m \ge n$$. The corresponding stable value of homology group is termed the stable homology group of degree $$q$$ for the symmetric groups.