# Integral cohomology of elementary abelian groups

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!

• Couldn't you do this using the Künneth formula for the cohomology of a direct product? – Derek Holt Dec 21 '15 at 17:56
• 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. – Dmitri Nikshych Dec 21 '15 at 18:40