Profinite group

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.

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.

Conjunction with other properties
Conjunction with other group properties: