Pro-finite 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.

1 Definition
2 Examples
3 Properties and facts
4 Pro-finite completion
5 Ind-finite groups
6 See also
7 References

[edit] Definition

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

[edit] 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 Z_p.

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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] See also

[edit] 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.