factor system

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

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

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 $G$.