# Profinite group

From Groupprops

## Contents

## Definition

### As a topological group

A **profinite group** is a topological group defined in the following equivalent ways:

- It is the inverse limit of an inverse system of finite groups, viewed as topological groups with the discrete topology.
- It is a compact Hausdorff totally disconnected topological group.

### As an abstract group

A **profinite group** is a group that arises as the inverse limit of an inverse system of finite groups. Note that profinite groups are usually studied along with their topologies and not as abstract groups; however, given an abstract group, we may be interested in whether it can be given any profinite group structure at all.

### Equivalence of definitions

`Further information: Equivalence of definition of profinite group`

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finite group|Find other variations of finite group |

## Examples

- The additive group of p-adic integers is an example of a profinite group. In fact, it is a pro-p-group.
- The profinite completion of the integers is an example of a profinite group.
- A direct product of (possibly infinitely many) finite groups has a natural structure as a profinite group if we take the product topology from the discrete topology on each of the factors.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Finite group | finite implies profinite | profinite not implies finite | Finitely generated profinite group|FULL LIST, MORE INFO | |

Direct product of finite groups | external direct product (unrestricted) of (possibly infinite, possibly repeated) finite groups | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Residually finite group | every non-identity element is outside a normal subgroup of finite index | profinite implies residually finite | residually finite not implies profinite | |FULL LIST, MORE INFO |

### Conjunction with other properties

Conjunction with other group properties:

Conjunction | Other component of conjunction | Intermediate notions |
---|---|---|

Finitely generated profinite group | Finitely generated group | |FULL LIST, MORE INFO |

Solvable profinite group | Solvable group | |FULL LIST, MORE INFO |

Abelian profinite group | Abelian group | |FULL LIST, MORE INFO |