Profinite group

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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

Definition

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space. 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.

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/pnZZ/pmZ (nm) 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 F2F1. 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.)

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