Editing Profinite group (section)

===Equivalence===

Any group constructed by the first definition satisfies the axioms in the second definition.

Conversely, any group <math>G</math> satisfying the axioms in the second definition can be constructed as an inverse limit according to the first definition using the inverse limit <math>\varprojlim G/N</math> where <math>N</math> ranges through the open [[normal subgroup]]s of <math>G</math> ordered by (reverse) inclusion. If <math>G</math> is topologically finitely generated then it is in addition equal to its own profinite completion.<ref>{{cite journal | last1=Nikolov| first1=Nikolay | last2=Segal| first2=Dan | title=On finitely generated profinite groups. I: Strong completeness and uniform bounds. II: Products in quasisimple groups | zbl=1126.20018 | journal=Ann. Math. |series=Second series | volume=165 | issue=1 | pages=171–238, 239–273 |date=2007 | doi=10.4007/annals.2007.165.171 | s2cid=15670650 | arxiv=math/0604399 }}</ref>