Editing Profinite group (section)

===Profinite completion===

Given an arbitrary group <math>G</math>, there is a related profinite group <math>\widehat{G},</math> the {{em|{{visible anchor|profinite completion}}}} of <math>G</math>.<ref name=":0" /> It is defined as the inverse limit of the groups <math>G/N</math>, where <math>N</math> runs through the [[normal subgroup]]s in <math>G</math> of finite [[Index of a subgroup|index]] (these normal subgroups are [[partial order|partially ordered]] by inclusion, which translates into an inverse system of natural homomorphisms between the quotients).

There is a natural homomorphism <math>\eta : G \to \widehat{G}</math>, and the image of <math>G</math> under this homomorphism is [[dense set|dense]] in <math>\widehat{G}</math>. The homomorphism <math>\eta</math> is injective if and only if the group <math>G</math> is [[residually finite group|residually finite]] (i.e.,
<math>\bigcap N = 1</math>, where the intersection runs through all normal subgroups <math>N</math> of finite index).

The homomorphism <math>\eta</math> is characterized by the following [[universal property]]: given any profinite group <math>H</math> and any continuous group homomorphism <math>f : G \rightarrow H</math> where <math>G</math> is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique [[continuous function (topology)|continuous]] group homomorphism <math>g : \widehat{G} \rightarrow H</math> with <math>f = g \eta</math>.