# virtually abelian group

A group $G$ is *virtually abelian* (or *abelian-by-finite*)
if it has an abelian^{} subgroup^{} (http://planetmath.org/Subgroup) of finite index (http://planetmath.org/Coset).

More generally, let $\chi $ be a property of groups.
A group $G$ is *virtually $\chi $* if it has a subgroup of finite index with the property $\chi $.
A group $G$ is *$\chi $-by-finite* if it has a normal subgroup^{} of finite index with the property $\chi $.
Note that every $\chi $-by-finite group is virtually $\chi $,
and the converse also holds if the property $\chi $ is inherited by subgroups.

These notions are obviously only of relevance to infinite groups, as all finite groups^{} are virtually trivial (and trivial-by-finite).

Title | virtually abelian group |

Canonical name | VirtuallyAbelianGroup |

Date of creation | 2013-03-22 14:35:58 |

Last modified on | 2013-03-22 14:35:58 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 11 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20F99 |

Classification | msc 20E99 |

Synonym | abelian-by-finite group |

Synonym | virtually-abelian group |

Related topic | VirtuallyCyclicGroup |

Defines | virtually abelian |

Defines | abelian-by-finite |

Defines | virtually nilpotent |

Defines | virtually solvable |

Defines | virtually polycyclic |

Defines | virtually free |

Defines | nilpotent-by-finite |

Defines | polycyclic-by-finite |

Defines | virtually nilpotent group |

Defines | virtually solvable group |

Defines | virtually polycyclic group |

Defines | virtually free |