Profinite completion

From Groupprops
Jump to: navigation, search

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.