# Profinite completion

From Groupprops

## Definition

Let be a group. The **profinite completion** of is the inverse limit of the inverse system of all finite quotients of , with maps as follows: for normal subgroups of with , there is a map given via the quotient map by (cf: third isomorphism theorem).

Note that itself maps to the inverse system of its finite quotients, so this gives rise to a natural homomorphism from to its profinite completion.

## Related properties

- For a finite group, the natural map to its profinite completion is an isomorphism.
- A group is a residually finite group if and only if the natural map to its profinite completion is injective.
- Any group arising as the profinite completion of some group is a profinite group.
- For a profinite group, the natural map to its profinite completion under the
*discrete*topology (rather than under the profinite topology) is an isomorphism.