Let $G$ be a finite group, then $H^n(G,\mathbb{Z})\neq 0$ for infinitely many $n$. This is not hard to see for cyclic groups. Can we prove this fact algebraically, could anyone provide a reference? Thanks
2026-03-29 03:53:23.1774756403
Non vanishing of group cohomology
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This link may prove useful:
https://mathoverflow.net/questions/64688/non-vanishing-of-group-cohomology-in-sufficiently-high-degree
As mentioned on the above page, Swan proved the result in:
R. G. Swan, The nontriviality of the restriction map in the cohomology of groups, Proc. Amer. Math. Soc. 11 (1960), 885-887.