Profinite group: Difference between revisions
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===As a topological group=== | ===As a topological group=== | ||
A '''profinite group''' is a 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 group]]s, viewed as [[topological group]]s with the discrete topology. | |||
# It is a [[compact group|compact]] [[T0 topological group|Hausdorff]] [[topospaces:totally disconnected space|totally disconnected]] [[topological group]]. | |||
{{group property}} | {{group property}} | ||
{{variation of|finite group}} | {{variation of|finite group}} | ||
Revision as of 21:05, 12 January 2012
Definition
As an abstract group
A profinite group is a group that arises as the inverse limit of an inverse system of finite groups.
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.
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 | |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 |