# Profinite group

In mathematics, **pro-finite groups** are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups.

## Contents

## Definition

Formally, a pro-finite group is a group that is isomorphic to the inverse limit of an inverse system of finite groups. Pro-finite groups are naturally regarded as topological groups: each of the finite groups carries the discrete topology, and since the inverse limit is a subset of the product of these discrete spaces, it inherits a topology which turns it into a topological group.

Equivalently, one can define pro-finite groups to be the topological groups that are Hausdorff, compact and totally disconnected.

## Examples

- Finite groups are pro-finite, if given the discrete topology.

- The group of
*p*-adic integers**Z**_{p under addition is pro-finite. It is the inverse limit of the finite groups Z/pnZ where n ranges over all natural numbers and the natural maps Z/pnZ → Z/pmZ (n≥m) are used for the limit process. The topology on this pro-finite group is the same as the topology arising from the p-adic valuation on Zp.}

- The Galois theory of field extensions of infinite degree gives rise naturally to Galois groups that are pro-finite. Specifically, if
*L*/*K*is a Galois extension, we consider the group*G*= Gal(*L*/*K*) consisting of all field automorphisms of*L*which keep all elements of*K*fixed. This group is the inverse limit of the finite groups Gal(*F*/*K*), where*F*ranges over all intermediate fields such that*F*/*K*is a*finite*Galois extension. For the limit process, we use the restriction homomorphisms Gal(*F*_{1}/*K*) → Gal(*F*_{2}/*K*), where*F*_{2}⊆*F*_{1}. The topology we obtain on Gal(*L*/*K*) is known as the**Krull topology**after Wolfgang Krull. Interestingly, Waterhouse showed that*every*pro-finite group is isomorphic to one arising from the Galois theory of*some*field*K*, but one cannot control which field*K*will be in this case. In fact, given a fixed field*K*, one does not know in general if even the finite groups occur as Galois groups over*K*. This is the Inverse Galois Problem for a field*K*.

- The fundamental groups considered in algebraic geometry are also pro-finite 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 pro-finite.

## Properties and facts

Every product of (arbitrarily many) pro-finite groups is pro-finite; the topology arising from the pro-finiteness agrees with the product topology.
Every closed subgroup of a pro-finite group is itself pro-finite; the topology arising from the pro-finiteness agrees with the subspace topology. If *N* is a closed normal subgroup of a pro-finite group *G*, then the factor group *G*/*N* is pro-finite; the topology arising from the pro-finiteness agrees with the quotient topology.

Since every pro-finite group *G* is compact Hausdorff, we have a Haar measure on *G*, which allows us to measure the "size" of subsets of *G*, compute certain probabilities, and integrate functions on *G*.

## Pro-finite completion

Given an arbitrary group *G*, there is a related pro-finite group *G*^{^}, the **pro-finite 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 them). 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. iff for every non-identity element *g* in *G* there exists a normal subgroup *N* in *G* of finite index that doesn't contain *g*). The homomorphism η is characterized by the following universal property: given any pro-finite group *H* and any group homomorphism *f* : *G* → *H*, there exists a unique continuous group homomorphism *g* : *G*^{^} → *H* with *f* = *g*η.

## Ind-finite groups

There is a notion of **ind-finite group**, which is the concept dual to pro-finite groups; i.e. a group *G* is ind-finite if it is the direct limit of finite groups. 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 pro-finite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian torsion groups.

See also: locally cyclic group.

## Further reading

- Hendrik Lenstra: Profinite Groups, talk given in Oberwolfach, November 2003. online version.
- Alexander Lubotzky: review of several books about pro-finite groups. Bulletin of the American Mathematical Society, 38 (2001), pages 475-479. online version.
- William C. Waterhouse.
*Profinite groups are Galois groups*. Proc. Amer. Math. Soc. 42 (1973), pp. 639–640.