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Pro-finite 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.

Examples

• Any product of finite groups is pro-finite.
• The p-adic integers Zp 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/pnZZ/pmZ (nm) 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 F2F1. The topology we obtain on Gal(L/K) is known as the Krull topology after Wolfgang Krull (1899 - 1971). Interestingly, every pro-finite group is isomorphic to one arising from Galois theory.

Properties and facts

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; the direct limit 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.

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.

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 : 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. 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 : GH, there exists a unique continuous group homomorphism g : G^H with f = gη.

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.

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.