Profinite 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 : G → G^, 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 : G → H, there exists a unique continuous group homomorphism g : G^ → H with f = gη.
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