# profinite completion

The *profinite completion* of a group $G$
is defined to be the profinite group

$$\widehat{G}={\underleftarrow{\mathrm{lim}}}_{N{\u22b4}_{\mathrm{f}}G}G/N,$$ |

where $N{\u22b4}_{\mathrm{f}}G$ means that $N$ is a normal subgroup^{} of finite index in $G$.

A group embeds into its profinite completion if and only if it is residually finite.

Title | profinite completion |
---|---|

Canonical name | ProfiniteCompletion |

Date of creation | 2013-03-22 15:15:54 |

Last modified on | 2013-03-22 15:15:54 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 10 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20E18 |

Related topic | AGroupsEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite |