Profinite group

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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.

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

  • The group of p-adic integers Zp 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/pnZZ/pmZ (nm) 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(F1/K) → Gal(F2/K), where F2F1. 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.

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 : 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.

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.

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