factor system
FactorSystem
2004-06-04 16:26:32
2004-06-04 16:26:32
Definition
group cohomology
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Using the notation in the entry Group Cohomology, a 2-cocycle or \emph{factor system} is a map $\phi:G\times G\ra M$ such that for all $\alpha,\beta,\gamma\in G$, we have
\begin{align*}
\alpha\phi(\beta,\gamma)-\phi(\alpha\beta,\gamma)+\phi(\alpha,\beta\gamma)-\phi(\beta,\gamma)=0.
\end{align*}
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 $G$.