Editing Profinite group (section)

==Procyclic group==

A profinite group <math>G</math> is {{em|{{visible anchor|procyclic group|procyclic|text=procyclic}}}} if it is topologically generated by a single element <math>\sigma;</math> that is, if <math>G = \overline{\langle \sigma \rangle},</math> the closure of the subgroup <math>\langle \sigma \rangle = \left\{\sigma^n: n \in \Z\right\}.</math><ref>{{Cite book|last=Neukirch|first=Jürgen|url=http://link.springer.com/10.1007/978-3-662-03983-0|title=Algebraic Number Theory|date=1999|publisher=Springer Berlin Heidelberg|isbn=978-3-642-08473-7|series=Grundlehren der mathematischen Wissenschaften|volume=322|location=Berlin, Heidelberg|doi=10.1007/978-3-662-03983-0}}</ref>

A topological group <math>G</math> is procyclic if and only if <math>G \cong {\textstyle\prod\limits_{p\in S}} G_p</math> where <math>p</math> ranges over some set of [[prime number]]s <math>S</math> and <math>G_p</math> is isomorphic to either <math>\Z_p</math> or <math>\Z/p^n \Z, n \in \N.</math><ref>{{Cite web|url=https://mathoverflow.net/questions/247731/decomposition-of-procyclic-groups|title=MO. decomposition of procyclic groups|website=MathOverflow}}</ref>