In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.

such that every group in the system can be generated 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 groupss 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 an indexed family each having the discrete topology, and a family of homomorphisms is the identity map 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

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

Profinite completion

the . runs through the normal subgroups of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients).

under this homomorphism is dense is residually finite of finite index).

is characterized by the following universal property is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique continuous.

Equivalence

Any group constructed by the first definition satisfies the axioms in the second definition.

ranges through the open normal subgroups is topologically finitely generated then it is in addition equal to its own profinite completion.

Surjective systemsSurjective system

In practice, the inverse system of finite groups is almost always and then it as its own profinite completion.

Examples

• Finite groups are profinite, if given the discrete topology.

• The group of p-adic integers under addition is profinite (in fact procyclic ranges over all natural numbers

• The group of profinite integers and it is the absolute Galois group of any finite field.

• The Galois theory of field extensions is a Galois extension consisting of all field automorphisms is a is known as the Krull topology after Wolfgang Krull. showed that profinite group is isomorphic to one arising from the Galois theory of one does not know in general precisely which finite groups This is the inverse Galois problem for a field the inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.) Not every profinite group occurs as an absolute Galois group of a field.

• The étale fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety. The fundamental groups of algebraic topology, however, are in general not profinite: for any prescribed group, there is a 2-dimensional CW complex whose fundamental group equals it.

• The automorphism group of a locally finite rooted tree is profinite.

Properties and facts

• Every product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the product topology. The inverse limit of an inverse system of profinite groups with continuous transition maps is profinite and the inverse limit functor is exact on the category of profinite groups. Further, being profinite is an extension property.

• Every closed subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the subspace topology then the factor group is profinite; the topology arising from the profiniteness agrees with the quotient topology.

• is compact Hausdorff, there exists a Haar measure compute certain probabilities, and integrate

• A subgroup of a profinite group is open if and only if it is closed and has finite index.

• According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely generated profinite group (that is, a profinite group that has a dense finitely generated subgroup) the subgroups of finite index are open. This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely generated p groupss. The proof uses the classification of finite simple groups.

• As an easy corollary of the Nikolov–Segal result above, is also of finite index, and hence it must be open.

• is a homeomorphism. Therefore the topology on a topologically finitely generated profinite group is uniquely determined by its structure.

Ind-finite groups

There is a notion of , which is the conceptual dual is ind-finite if it is the direct limit of an inductive system of finite groups. (In particular, it is an ind-group 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 there is a section

is equivalent to either of the two properties:

• the cohomological dimension

• -groups.

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

is

ranges over some set of prime numbers

See also

References

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