cohomology of permutation group with mod 2 coefficient

Question

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$

For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I know the answer.

The papers are very difficult. For example, what is $$ H^*(BS_3;\mathbb{Z}_2)? $$

Answer 1

This is one of the places where generalizing makes structure more visible. If you are willing to consider $S_n$ for all $n$ at once, i.e., study $$ \coprod_n BS_n, $$ then the cohomology algebra has two products making it a Hopf ring (a ring object in the category of coalgebras).

Giusti, Salvatore, and Sinha studied this and include explicit rules for multiplication in terms of combinatorial objects called skyline diagrams (reminiscent of Young diagrams, not surprisingly).

---

EDIT:

Have a look at this more recent paper by Giusti and Sinha that is more easily digestible and goes into considerably more detail. Also, there's the book Cohomology of Finite Groups by Adem and Milgram.