 Group cohomology

For a group $G$ acting on an abelian group $M$, group cohomology consists of the groups $H^n(G,M)$ obtained from the standard cochain complex of maps $G^n\to M$. In degree zero it gives the invariant subgroup $M^G$. In degree one it gives one-cocycles modulo group coboundaries.

If the action is trivial, the one-cocycle identity becomes $f(gh)=f(g)+f(h)$ and all one-coboundaries vanish, so $H^1(G,M)=\mathrm{Hom}(G,M)$. Degree two encodes group extensions with the prescribed action on the kernel. This explains why cohomology naturally measures whether local multiplication data can be assembled into a globally associative extension.

Group cohomology is a mathematical tool used in algebraic topology, group theory, and various other areas of mathematics. It provides a way to study the properties of groups using cohomological methods, which are analogous to those used in homology theory but focus on the algebraic structure associated with groups.