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In mathematics, group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group G in an associated G-module M to elucidate the properties of the group.
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one-cocycles over finite groups
Let $G$ be a finite group and let $S$ be a subgroup of $G$. Let
$\alpha$ be a one-cocycle $G\to M$ for some $G$-module $M$, and
suppose that $Res_{S}^{G}\alpha=0$. For $g\in G$, and $S^{g}=gSg^{-1}$,
…