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.
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For a group acting on an abelian group , group cohomology consists of the groups obtained from the standard cochain complex of maps . In degree zero it gives the invariant subgroup . In degree one it gives one-cocycles modulo group coboundaries.
If the action is trivial, the one-cocycle identity becomes and all one-coboundaries vanish, so . 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.