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 Hausdorff compact and totally disconnected topological group. Equivalently, one can define a pro-finite groups as topological groups isomorphic to inverse limits (in the category of topological groups) of an inverse (aka, projective) system of finite groups, regarded as discrete topological groups.

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/p^mZ (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(F₁/K) → Gal(F₂/K), where F₂ ⊆ F₁. The topology we obtain on Gal(L/K) is known as the Krull topology after Wolfgang Krull. Waterhouse showed that every pro-finite group is isomorphic to one arising from the Galois theory of some field K; but one cannot (yet) control which field K will be in this case. In fact, for many fields K one does not know in general precisely which finite groups occur as Galois groups over K. This is the Inverse Galois problem for a field K. (For some fields K the Inverse Galois problem is settled, such as the field of rational functions in one variable over the complex numbers.)

• 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.
• Any open subgroup has finite index, and a closed subgroup is open if and only if it has finite index.
• According to a theorem of Nikolay Nikolov and Dan Segal, in any topologically finitely-generated profinite group the subgroups of finite index are open. This generalizes an earlier analogous result of Jean-Pierre Serre for pro-p groups. The proof uses the classification of finite simple groups.
• As an easy corollary of the Nikolov-Segal result above, any surjective discrete group homomorphism φ: G → H between profinite groups G and H is continuous as long as G is topologically finitely-generated. Indeed, any open set of H is of finite index, so its preimage in G is also of finite index, hence it must be open.
• Suppose G and H are topologically finitely-generated profinite groups which are isomorphic as discrete groups by an isomorphism ι. Then ι is bijective and continuous by the above result. Furthermore, ι⁻¹ is also continuous, so ι is a homeomorphism. Therefore the topology on a topologically finitely-generated profinite group is uniquely determined by its algebraic structure.

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. if and only if 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 = 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 an inductive system 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

References

• Nikolay Nikolov and Dan Segal. On finitely generated profinite groups I: strong completeness and uniform bounds.. 2006, online version.
• Nikolay Nikolov and Dan Segal. On finitely generated profinite groups II, products in quasisimple groups. 2006, online version.
• 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.
• J. P. Serre, Cohomologie Galoisienne. Springer Lecture Notes in Mathematics, vol. 5.
• William C. Waterhouse. Profinite groups are Galois groups. Proc. Amer. Math. Soc. 42 (1973), pp. 639–640.

Group theory | Topological groups

Krulltopologie | Grupo profinito