# factor system

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 $G$ by $M$ in the case where $M$ is taken to be an abelian normal subgroup of $G$.

Title factor system FactorSystem 2013-03-22 14:23:28 2013-03-22 14:23:28 mathcam (2727) mathcam (2727) 4 mathcam (2727) Definition msc 20J06