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.


Properties and facts

Profinite completion

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.

Ind-finite groups

There is a notion of ind-finite group, which is the concept dual to profinite groups; i.e. a group G is ind-finite if it is the direct limit of an inductive system of finite groups. (In particular, it is an ind-group.) The usual terminology is different: a group G is called locally finite if every finitely-generated subgroup is finite. This is equivalent, in fact, to being 'ind-finite'.

By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian torsion groups.

Projective profinite groups

A profinite group is projective if it has the lifting property for every extension. This is equivalent to saying that G is projective if for every surjective morphism from a profinite HG there is a section GH.[2][3]

Projectivity for a profinite group G is equivalent to either of the two properties:[2]

Every projective profinite group can be realized as an absolute Galois group of a pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries.[4]

See also


  1. Fried & Jarden (2008) p. 497
  2. 1 2 Serre (1997) p. 58
  3. Fried & Jarden (2008) p. 207
  4. Fried & Jarden (2008) pp. 208,545
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