# Profinite group

## Definition

### As a topological group

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

1. It is the inverse limit of an inverse system of finite groups, viewed as topological groups with the discrete topology.
2. 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 properties
VIEW 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 |

## 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