# 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 $G$.

Title | factor system |
---|---|

Canonical name | FactorSystem |

Date of creation | 2013-03-22 14:23:28 |

Last modified on | 2013-03-22 14:23:28 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 4 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 20J06 |