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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Group cohomology studies a group through cochain complexes built from its actions on abelian groups. In degree one, identifies one-cocycles modulo one-coboundaries.