Pro-finite group

In mathematics, a pro-finite group G is a group that, in a sense, is very "close" to being finite.

Table of contents

1 Definition
2 Examples
3 Properties
4 Ind-finite groups

Definition

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

Examples

Every finite group is pro-finite, but that is boring.
The p-adic integers Z_{p are pro-finite (with respect to addition): they are the inverse limit of the finite groups Z/pⁿZ where n ranges over all natural numbers and the natural maps Z/pⁿZ → Z/p^mZ (n≥m) are used for the limit process.}
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(F₁/K) → Gal(F₂/K), where F₂ ⊆ F₁.
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 are in general not pro-finite.)

Properties

Every pro-finite group is a compact Hausdorff space: since all finite discrete spaces are compact Hausdorff spaces, their product will be a compact Hausdorff space by Tychonoff's theorem. G is a closed subset of this product and is therefore also compact Hausdorff.

Every pro-finite group is totally disconnected and even more: a topological group is pro-finite if and only if it is Hausdorff, compact and totally disconnected.

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.