# Profinite completion

## Definition

Let $G$ be a group. The profinite completion of $G$ is the inverse limit of the inverse system of all finite quotients of $G$, with maps as follows: for normal subgroups $N_1, N_2$ of $G$ with $N_2 \le N_1$, there is a map $G/N_2 \to G/N_1$ given via the quotient map by $N_1/N_2$ (cf: third isomorphism theorem).

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