# Profinite completion of the integers

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Contents

## Definition

The **profinite completion of the integers** is defined in the following equivalent ways. Note that these definitions can be interpreted both as defining the group abstractly and as defining the group as a topological group:

- The group obtained by taking the profinite completion of the group of integers (viewed as a discrete group).
- The external direct product, over all prime numbers , of the additive group of p-adic integers for the prime . Note that we take the product topology from the topologies on these groups.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order of a profinite group | for all primes . | This is a supernatural number as is the convention for orders of profinite groups. |

## Group properties

### Abstract group properties

Property | Satisfied? | Explanation |
---|---|---|

abelian group | ||

aperiodic group |

### Topological group properties

Here, the topology is from the profinite group structure.

Property | Satisfied? | Explanation |
---|---|---|

profinite group | Yes | By definition |

compact group | Yes | profinite groups are compact |

T0 topological group | Yes | profinite groups are Hausdorff, hence |

totally disconnected group | Yes | profinite groups are totally disconnected |

topologically cyclic group | Yes | , the subgroup generated by the element that projects to 1 mod for all , is a dense subgroup. In fact, this group can be thought of as a compactification of . |

topologically finitely generated group | Yes | Follows from being topologically cyclic. |

connected topological group | No | It is nontrivial and totally disconnected. |