Editing Profinite group (section)

==Examples==

* Finite groups are profinite, if given the [[discrete topology]].
* The group of [[p-adic number|<math>p</math>-adic integers]] <math>\Z_p</math> under addition is profinite (in fact [[#Procyclic group|procyclic]]). It is the inverse limit of the finite groups <math>\Z/p^n\Z</math> where <math>n</math> ranges over all [[natural number]]s and the natural maps <math>\Z/p^n\Z \to \Z/p^m\Z</math> for <math>n \ge m.</math> The topology on this profinite group is the same as the topology arising from the <math>p</math>-adic valuation on <math>\Z_p.</math>
* The group of [[profinite integer]]s <math>\widehat{\Z}</math> is the profinite completion of <math>\Z.</math> In detail, it is the inverse limit of the finite groups <math>\Z/n\Z</math> where <math>n = 1,2,3,\dots</math> with the modulo maps <math>\Z/n\Z \to \Z/m\Z</math> for <math>m\,|\,n.</math> This group is the product of all the groups <math>\Z_p,</math> and it is the [[absolute Galois group]] of any [[finite field]].
* The [[Galois theory]] of [[field extension]]s of infinite degree gives rise naturally to Galois groups that are profinite. Specifically, if <math>L / K</math> is a [[Galois extension]], consider the group <math>G = \operatorname{Gal}(L / K)</math> consisting of all [[field automorphism]]s of <math>L</math> that keep all elements of <math>K</math> fixed. This group is the inverse limit of the finite groups <math>\operatorname{Gal}(F / K),</math> where <math>F</math> ranges over all intermediate fields such that <math>F / K</math> is a {{em|finite}} Galois extension. For the limit process, the restriction homomorphisms <math>\operatorname{Gal}(F_1 / K) \to \operatorname{Gal}(F_2 / K)</math> are used, where <math>F_2 \subseteq F_1.</math> The topology obtained on <math>\operatorname{Gal}(L / K)</math> is known as the ''[[Krull topology]]'' after [[Wolfgang Krull]]. {{harvtxt|Waterhouse|1974}} showed that {{em|every}} profinite group is isomorphic to one arising from the Galois theory of {{em|some}} field <math>K,</math> but one cannot (yet) control which field <math>K</math> will be in this case.  In fact, for many fields <math>K</math> one does not know in general precisely which [[finite group]]s occur as Galois groups over <math>K.</math> This is the [[inverse Galois problem]] for a field&nbsp;<math>K.</math> (For some fields <math>K</math> the inverse Galois problem is settled, such as the field of [[rational function]]s in one variable over the complex numbers.)  Not every profinite group occurs as an [[absolute Galois group]] of a field.<ref name=FJ497>Fried & Jarden (2008) p.&nbsp;497</ref>
* The [[Étale fundamental group|étale 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 group]]s of [[algebraic topology]], however, are in general not profinite: for any prescribed group, there is a 2-dimensional [[CW complex]] whose fundamental group equals it.
* The automorphism group of a [[locally finite rooted tree]] is profinite.