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Group homology

In abstract algebra, one uses homology to define derived functors, for example the Tor functors. 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 F 1 {\displaystyle F_{1}} and a surjective homomorphism p 1: F 1 → X. {\displaystyle p_{1}:F_{1}\to X.} Then one finds a free module F 2 {\displaystyle F_{2}} and a surjective homomorphism p 2: F 2 → ker (p 1). {\displaystyle p_{2}:F_{2}\to \ker \left(p_{1}\right).} Continuing in this fashion, a sequence of free modules F n {\displaystyle F_{n}} and homomorphisms p n {\displaystyle p_{n}} can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology H n {\displaystyle H_{n}} 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 H 2 (G, M) {\displaystyle H^{2}(G,M)} is to classify the possible extension groups E which contain a given G -module M as a normal subgroup and have a given quotient group G, so that G = E / M. {\displaystyle G=E/M.}