Profinite group

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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

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