Profinite Group - Examples

Examples

• Finite groups are profinite, if given the discrete topology.

• The group of p-adic integers Z_p 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 Z_p.

• 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 (1974) 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.) Not every profinite group occurs as an absolute Galois group of a field.

• The fundamental groups considered in algebraic geometry are also profinite 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 profinite.

• The automorphism group of a locally finite rooted tree is profinite.