The homology groups of every finite group are finite and those of $\mathbb{Z}^n$ with coefficients in $\mathbb{Z}$ are finitely generated. I was wondering whether for some finitely generated group there is a homology group $H^n(G,\mathbb{Z})$ for some $n\in\mathbb{N}$ which is not finitely generated. This of course can not take place for any group $G$ whose classifying space is compact as the homology groups will be finitely generated in this case.
2026-03-26 12:36:46.1774528606
homology of finitely generated group
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