# abelianization

The *abelianization ^{}* of a group $G$ is $G/[G,G]$, the quotient (http://planetmath.org/QuotientGroup) of $G$ by its derived subgroup.

The abelianization of $G$ is the largest abelian^{} quotient of $G$, in the sense that if $N$ is a normal subgroup^{} of $G$ then $G/N$ is abelian if and only if $[G,G]\subseteq N$.
In particular, every abelian quotient of $G$ is a homomorphic image^{} of $G/[G,G]$.

If $A$ is an abelian group and $\varphi :G\to A$ is a homomorphism^{} (http://planetmath.org/GroupHomomorphism),
then there is a unique homomorphism $\psi :G/[G,G]\to A$ such that
$\psi \circ \pi =\varphi $, where $\pi :G\to G/[G,G]$ is the canonical projection.

Title | abelianization |
---|---|

Canonical name | Abelianization |

Date of creation | 2013-03-22 14:52:57 |

Last modified on | 2013-03-22 14:52:57 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 7 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20F14 |

Synonym | abelianisation |

Related topic | DerivedSubgroup |