Group with periodic cohomology

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Definition

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.

A finite group with periodic cohomology is a finite group that has periodic cohomology.