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Let $p$ be a prime. I am looking for a reference or a short proof for the fact that cohomology groups $H^i((\mathbb{Z}/p\mathbb{Z})^n,\, \mathbb{Z}),\, i>0,$ have exponent $p$ (i.e., that they are vector spaces over the field $\mathbb{Z}/p\mathbb{Z}$). Also, I can explicitly describe these groups (at least for small $i$) as $GL_n(\mathbb{Z}/p\mathbb{Z})$-modules, but I suppose this is well known. Will appreciate a reference!

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    $\begingroup$ Couldn't you do this using the Künneth formula for the cohomology of a direct product? $\endgroup$
    – Derek Holt
    Commented Dec 21, 2015 at 17:56
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    $\begingroup$ Thanks! Indeed, one proceeds inductively: this is clearly true for $n=1$, the exact sequence in the Künneth formula splits, and all $Tor$ groups are elementary Abelian in this case. $\endgroup$ Commented Dec 21, 2015 at 18:40

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