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 |

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	|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