Editing Homology (mathematics) (section)

=== Group homology ===
{{Main|Group cohomology}}

In [[abstract algebra]], one uses homology to define [[derived functor]]s, for example the [[Tor functor]]s. Here one starts with some covariant additive functor ''F'' and some module ''X''. The chain complex for ''X'' is defined as follows: first find a free module <math>F_1</math> and a [[surjective]] homomorphism <math>p_1 : F_1 \to X.</math> Then one finds a free module <math>F_2</math> and a surjective homomorphism <math>p_2 : F_2 \to \ker\left(p_1\right).</math> Continuing in this fashion, a sequence of free modules <math>F_n</math> and homomorphisms <math>p_n</math> can be defined. By applying the functor ''F'' to this sequence, one obtains a chain complex; the homology <math>H_n</math> of this complex depends only on ''F'' and ''X'' and is, by definition, the ''n''-th derived functor of ''F'', applied to ''X''.

A common use of group (co)homology <math>H^2(G, M)</math> is to classify the possible [[Group extension|extension groups]] ''E'' which contain a given ''G''-module ''M'' as a [[normal subgroup]] and have a given [[quotient group]] ''G'', so that <math>G = E / M.</math>