I've had trouble finding a reference.
I know that the classifying space of a group $G$ is an example of an Eilenberg-MacLane space $K(G,1)$, so that the cohomology of $K(G,1)$ is group cohomology. If $G$ is finite, group cohomology is killed by the order $|G|$ of $G$.
I believe it's no longer true that $|G|$ kills $H^*(K(G,n),\mathbb{Z})$ for $n > 1$. For example, the tables in the appendix of Alain Clement's thesis seem to give examples of homology groups of $K(\mathbb{Z}/2,2)$ with elements of order $2^k$ for $k > 1$. However, all these groups seem to have order a power of $2$.
My question is: are the (co)homology groups of $K(G,n)$ always $p$-groups when $G$ is a finite $p$-group? Is there an easy way to see this?
Thanks!
Best Answer
A bit more elementary than using localizations of spaces, you can prove this by induction on $n$ using the Serre spectral sequence. Specifically, consider the Serre spectral sequence for homology with coefficients in $\mathbb{Z}[1/p]$ for the fiber sequence $K(G,n)\to *\to K(G,n+1)$. This tells you that if $n\geq 1$ and $K(G,n)$ has trivial reduced homology with coefficients in $\mathbb{Z}[1/p]$ then so does $K(G,n+1)$. Starting from the base case $n=1$, it then follows that $\tilde{H}_*(K(G,n),\mathbb{Z}[1/p])\cong \tilde{H}_*(K(G,n),\mathbb{Z})\otimes\mathbb{Z}[1/p]$ is trivial for all $n\geq 1$.