Profinite group: Difference between revisions
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==Definition== | ==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 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]]. | |||
===As an abstract group=== | ===As an abstract group=== | ||
A '''profinite group''' is a group that arises as the [[inverse limit]] of an inverse system of [[finite group]]s. | A '''profinite group''' is a group that arises as the [[inverse limit]] of an inverse system of [[finite group]]s. 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|[[Equivalence of definition of profinite group]]}} | |||
{{group property}} | {{group property}} | ||
{{variation of|finite group}} | {{variation 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== | ==Relation with other properties== | ||
Latest revision as of 00:23, 15 January 2012
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 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 |