In mathematics (more specifically, in homological algebra), 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 G n {\displaystyle G^{n}} representing n-simplices, topological properties of the space may be computed, such as the set of cohomology groups H n ( G , M ) {\displaystyle H^{n}(G,M)} . 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. These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group G is the singular cohomology of a suitable space having G as its fundamental group, namely the corresponding Eilenberg–MacLane space. Thus, the group cohomology of Z {\displaystyle \mathbb {Z} } can be thought of as the singular cohomology of the circle S1. Likewise, the group cohomology of Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is the singular cohomology of infinite real projective space P ∞ ( R ) . {\displaystyle \mathbb {P} ^{\infty }(\mathbb {R} ).}
A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.
Motivation A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an automorphism of M. We will write G multiplicatively and M additively. Given such a G-module M, it is natural to consider the submodule of G-invariant elements:
M G = { x ∈ M | ∀ g ∈ G : g x = x } . {\displaystyle M^{G}=\lbrace x\in M\ |\ \forall g\in G:\ gx=x\rbrace .}
Now, if N is a G-submodule of M (i.e., a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in M / N {\displaystyle M/N} are found as the quotient of the invariants in M by those in N: being invariant 'modulo N ' is broader. The purpose of the first group cohomology H 1 ( G , N ) {\displaystyle H^{1}(G,N)} is to precisely measure this difference. The group cohomology functors H ∗ {\displaystyle H^{*}} in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence.
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