# Group with periodic cohomology

A group $G$ is said to have periodic cohomology if there exists a cohomology class $c \in H^d(G;\mathbb{Z})$ for some positive integer $d$, such that the cup product with $c$ defines an isomorphism between $H^m(G;\mathbb{Z})$ and $H^{m+d}(G;\mathbb{Z})$ for every $m$. In particular, the sequence of cohomology groups is periodic.