# Group cohomology

In

abstract algebra,

homological algebra,

algebraic topology and

algebraic number theory, as well as in applications to

group theory proper,

**group cohomology** arises as a sequence of functors

*H*^{n}. The

*H*^{0} functor can be described directly as the subgroup of G-invariant elements, for any

abelian group on which a

group G acts (by endomorphisms). Then the other functors, for

*n* = 1, 2, 3 ... arise because taking invariants doesn't respect exact sequences.

In more concrete terms, if B is a subgroup of A mapped to itself by the action of G, it isn't in general true that the invariants in A/B are found as the quotient of the invariants in A by the invariants in B: being invariant 'up to something in B' is broader. The difference then lies in *H*^{1}(B). That is, by general procedures (cf. derived functor) a long exact sequence is constructed.

Early recognition of group cohomology came in the *Noether's equations* of Galois theory (an appearance of cocycles for *H*^{1}), and the *factor sets* of the extension problem for groups (Schur's multiplicator) and in simple algebras (Brauer), both of these latter being connected with *H*^{2}. Some general theory was supplied by Mac Lane and Lyndon; from a module-theoretic point of view this was integrated into the Cartan-Eilenberg theory, and topologically into an aspect of the construction of the classifying space BG for G-bundles. The application to class field theory provided theorems for general Galois extensions (not just abelian extensions).

Some refinements in the theory post-1960 have been made (continuous cocycles, Tate's redefinition) but the basic outlines remain the same.

The analoguous theory for Lie algebras is formally similar, starting with the corresponding definition of *invariant*. It is much applied in representation theory.