Editing Profinite group (section)

==Projective profinite groups==

A profinite group is {{em|{{visible anchor|projective profinite groups|projective|text=projective}}}} if it has the [[lifting property]] for every extension.  This is equivalent to saying that <math>G</math> is projective if for every surjective morphism from a profinite <math>H \to G</math> there is a [[Section (category theory)|section]] <math>G \to H.</math><ref name=S9758>Serre (1997) p.&nbsp;58</ref><ref name=FJ207>Fried & Jarden (2008) p.&nbsp;207</ref>

Projectivity for a profinite group <math>G</math> is equivalent to either of the two properties:<ref name=S9758/>
* the [[cohomological dimension]] <math>\operatorname{cd}(G) \leq 1;</math>
* for every prime <math>p</math> the Sylow <math>p</math>-subgroups of <math>G</math> are free pro-<math>p</math>-groups.

Every projective profinite group can be realized as an [[absolute Galois group]] of a [[pseudo algebraically closed field]]. This result is due to [[Alexander Lubotzky]] and [[Lou van den Dries]].<ref>Fried & Jarden (2008) pp.&nbsp;208,545</ref>