Group with periodic cohomology

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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.