PlanetMath: factor system

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factor system

(Definition)

Using the notation in the entry Group Cohomology, a 2-cocycle or factor system is a map $\phi:G\times G\rightarrow M$ such that for all $\alpha,\beta,\gamma\in G$ , we have

$\displaystyle \alpha\phi(\beta,\gamma)-\phi(\alpha\beta,\gamma)+\phi(\alpha,\beta\gamma)-\phi(\beta,\gamma)=0.$

Factor systems play a role in determing classes of group extensions of by in the case where is taken to be an abelian normal subgroup of .

"factor system" is owned by mathcam.

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Cross-references: normal subgroup, abelian, group extensions, classes, map, group cohomology

This is version 1 of factor system, born on 2004-06-04.
Object id is 5887, canonical name is FactorSystem.
Accessed 1327 times total.

Classification:

AMS MSC:

20J06 (Group theory and generalizations :: Connections with homological algebra and category theory :: Cohomology of groups)

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