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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 $H_i(S_n)$?

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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.

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