group cohomology

Let G be a group and let M be a (left) G-module. The 0th cohomology groupPlanetmathPlanetmath of the G-module M is


which is the set of elements of M which are G-invariant, also denoted by MG.

A map ϕ:GM is said to be a crossed homomorphism (or 1-cocycle) if


for all α,βG. If we fix mM, the map ρ:GM defined by


is clearly a crossed homomorphism, said to be principal (or 1-coboundary). We define the following groups:

Z1(G,M) = {ϕ:GM:ϕ is a 1-cocycle}
B1(G,M) = {ρ:GM:ρ is a 1-coboundary}

Finally, the 1st cohomology group of the G-module M is defined to be the quotient groupMathworldPlanetmath:


The following proposition is very useful when trying to compute cohomology groups:

Proposition 1.

Let G be a group and let A,B,C be G-modules related by an exact sequencePlanetmathPlanetmathPlanetmath:


Then there is a long exact sequence in cohomology:


In general, the cohomology groups Hn(G,M) can be defined as follows:

Definition 1.

Define C0(G,M)=M and for n1 define the additive groupMathworldPlanetmath:


The elements of Cn(G,M) are called n-cochains. Also, for n0 define the nth coboundaryMathworldPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath dn:Cn(G,M)Cn+1(G,M):

dn(ϕ)(g1,,gn+1) = g1ϕ(g2,,gn+1)
+ i=1n(-1)iϕ(g1,,gi-1,gigi+1,gi+2,,gn+1)
+ (-1)n+1ϕ(g1,,gn)

Let Zn(G,M)=kerdn for n0, the set of n-cocyles. Also, let B0(G,M)=1 and for n1 let Bn(G,M)=imagedn-1, the set of n-coboundaries.

Finally we define the nth-cohomology group of G with coefficients in M to be



Title group cohomologyMathworldPlanetmath
Canonical name GroupCohomology
Date of creation 2013-03-22 13:50:07
Last modified on 2013-03-22 13:50:07
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 11
Author alozano (2414)
Entry type Definition
Classification msc 20J06
Synonym cohomology
Related topic SelmerGroup
Related topic CohomologyGroupTheorem
Related topic ProofOfCohomologyGroupTheorem
Related topic OmegaSpectrum
Related topic NaturalEquivalenceOfC_GAndC_MCategories
Defines group cohomology
Defines coboundary
Defines cocycle
Defines crossed homomorphism