Let $S\_n$ denote the symmetric group on $n$ letters, and let $S\_n(p)$ denote a Sylow $p$-subgroup. Why is the image of $H\_i(S\_n(p))$ in $H\_i(S\_n)$ the $p$-primary part of $H\_i(S\_n)$?



Homology of symmetric groups
That is the content of the first part of theorem 10.1 in Cartan-Eilenberg (up to the fact that their $\hat H^i$ is the same as $H\_{-i-1}$ for $i<-1$). In particular, this is true for all finite groups, not just the symmetric ones.


