Profinite group: Difference between revisions

→‎Examples: added example of profinite integers
(→‎Examples: added example of profinite integers)
* Finite groups are profinite, if given the [[discrete topology]].
* The group of [[p-adic number|''p''-adic integers]] <math>\Z_p</math> under addition is profinite (in fact [[procyclic (mathematics)|procyclic]]). It is the inverse limit of the finite groups <math>\Z/p^n\Z</math> where ''n'' ranges over all natural numbers and the natural maps <math>\Z/p^n\Z \to \Z/p^m\Z</math>(<math>n\ge m</math>) 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 <math>\Z_p</math>.
* The group of [[profinite integers]] <math>\widehat{\Z}</math> is the inverse limit of the finite groups <math>\Z/n\Z</math> where <math>n = 1,2,3,\dots</math> and we use the maps <math>\Z/n\Z \to \Z/m\Z</math> for <math>n|m</math> are used in the limit process. 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 ''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(''F''<sub>1</sub>/''K'') → Gal(''F''<sub>2</sub>/''K''), where ''F''<sub>2</sub> ⊆ ''F''<sub>1</sub>. The topology we obtain on Gal(''L''/''K'') is known as the '''Krull topology''' after [[Wolfgang Krull]]. {{harvtxt|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 group]]s occur as Galois groups over ''K''. This is the [[inverse Galois problem]] for a field&nbsp;''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.<ref name=FJ497>Fried & Jarden (2008) p.&nbsp;497</ref>
* The [[Étale fundamental group|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 (fix a presentation of the group; the CW complex has one 0-cell, a loop for every generator, and a 2-cell for every relation, whose attaching map corresponds to the relation in the "obvious" way: e.g. for the relation ''abc=1'', the attaching map traces a generator of the fundamental groups of the loops for ''a'', ''b'', and ''c'' in order. The computation follows by [[van Kampen's theorem]].)