*** Welcome to piglix ***

Pro-finite group


In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.

A non-compact generalization of a profinite group is a locally profinite group.

A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. Equivalently, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space. In categorical terms, this is a special case of a (co)filtered limit construction.

Given an arbitrary group G, there is a related profinite group G^, the profinite completion of G. It is defined as the inverse limit of the groups G/N, where N runs through the normal subgroups in G of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients). There is a natural homomorphism η : GG^, and the image of G under this homomorphism is dense in G^. The homomorphism η is injective if and only if the group G is residually finite (i.e., , where the intersection runs through all normal subgroups of finite index). The homomorphism η is characterized by the following universal property: given any profinite group H and any group homomorphism f : GH, there exists a unique continuous group homomorphism g : G^H with f = gη.


...
Wikipedia

...