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. By treating the
G-module as a kind of topological space with elements of
representing
n-
simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group
G and
G-module
M themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the
quotient module or space with respect to a group action. Group cohomology is used in the fields of
abstract algebra,
homological algebra,
algebraic topology and
algebraic number theory, as well as in applications to
group theory proper. As in algebraic topology, there is a dual theory called
group homology. The techniques of group cohomology can also be extended to the case that instead of a
G-module,
G acts on a nonabelian
G-group; in effect, a generalization of a module to
non-Abelian coefficients.