In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d\in\N such that every group in the system can be generated by d elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profinite group. Important examples of profinite groups are the additive groups of p-adic integers and the Galois groups of infinite-degree field extensions. Every profinite group is compact and totally disconnected. A non-compact generalization of the concept is that of locally profinite groups. Even more general are the totally disconnected groups.

Definition

Profinite groups can be defined in either of two equivalent ways.

First definition (constructive)

A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. In this context, an inverse system consists of a directed set (I, \leq), an indexed family of finite groups each having the discrete topology, and a family of homomorphisms such that f_i^i is the identity map on G_i and the collection satisfies the composition property whenever The inverse limit is the set: equipped with the relative product topology. One can also define the inverse limit in terms of a universal property. In categorical terms, this is a special case of a cofiltered limit construction.

Second definition (axiomatic)

A profinite group is a compact and totally disconnected topological group: that is, a topological group that is also a Stone space.

Profinite completion

Given an arbitrary group G, there is a related profinite group the 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, and the image of G under this homomorphism is dense in \widehat{G}. The homomorphism \eta is injective if and only if the group G is residually finite (i.e.,, where the intersection runs through all normal subgroups N of finite index). The homomorphism \eta is characterized by the following universal property: given any profinite group H and any continuous group homomorphism where G is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique continuous group homomorphism with f = g \eta.

Equivalence

Any group constructed by the first definition satisfies the axioms in the second definition. Conversely, any group G satisfying the axioms in the second definition can be constructed as an inverse limit according to the first definition using the inverse limit where N ranges through the open normal subgroups of G ordered by (reverse) inclusion. If G is topologically finitely generated then it is in addition equal to its own profinite completion.

Surjective systems

In practice, the inverse system of finite groups is almost always, meaning that all its maps are surjective. Without loss of generality, it suffices to consider only surjective systems since given any inverse system, it is possible to first construct its profinite group G, and then it as its own profinite completion.

Examples

Properties and facts

Ind-finite groups

There is a notion of, which is the conceptual 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 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 H \to G there is a section G \to H. Projectivity for a profinite group G is equivalent to either of the two properties: 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.

Procyclic group

A profinite group G is if it is topologically generated by a single element \sigma; that is, if the closure of the subgroup A topological group G is procyclic if and only if where p ranges over some set of prime numbers S and G_p is isomorphic to either \Z_p or