Profinite group

From Wikipedia, the free encyclopedia

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups; they share many properties with their finite quotients.

Contents

[edit] Definition

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group. Equivalently, one can define a profinite group to be a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. In categorical terms, this is a special case of a (co)filtered limit construction.

[edit] Examples

  • The group of p-adic integers Zp under addition is profinite (in fact procyclic). 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/pmZ (n≥m) are used for the limit process. The topology on this profinite 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 profinite. 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 profinite 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.)

[edit] Properties and facts

  • Every product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the product topology.
  • Every closed subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the subspace topology. If N is a closed normal subgroup of a profinite group G, then the factor group G/N is profinite; the topology arising from the profiniteness agrees with the quotient topology.
  • Since every profinite 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 topologically finitely-generated 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] Profinite completion

Given an arbitrary group G, there is a related profinite group G^, the profinite 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 the quotients). 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., \cap N = 1, where the intersection runs through all normal subgroups of finite index). The homomorphism η is characterized by the following universal property: given any profinite 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 profinite 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 profinite 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 profinite groups. Bulletin of the American Mathematical Society, 38 (2001), pages 475-479. online version.
  • Serre, Jean-Pierre (1994), Cohomologie galoisienne, vol. 5, Lecture Notes in Mathematics, Berlin, New York: Springer-Verlag, MR1324577, ISBN 978-3-540-58002-7 
  • William C. Waterhouse. Profinite groups are Galois groups. Proc. Amer. Math. Soc. 42 (1973), pp. 639–640.