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.