Editing Profinite group (section)

==Definition==

Profinite groups can be defined in either of two equivalent ways.

===First definition (constructive)===

A profinite group is a topological group that is [[isomorphism|isomorphic]] to the [[inverse limit]] of an [[inverse system]] of [[discrete space|discrete]] finite groups.<ref>{{Cite web|url=http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf|title=Profinite Groups|last=Lenstra|first=Hendrik|website=Leiden University}}</ref> In this context, an inverse system consists of a [[directed set]]  <math>(I, \leq),</math> an [[indexed family]] of finite groups <math>\{G_i: i \in I\},</math> each having the [[discrete topology]], and a family of [[Group homomorphism|homomorphisms]] <math>\{f^j_i : G_j \to G_i \mid i, j \in I, i \leq j\}</math> such that <math>f_i^i</math> is the [[identity map]] on <math>G_i</math> and the collection satisfies the composition property <math>f^j_i \circ f^k_j = f^k_i</math> whenever <math>i\leq j\leq k.</math> The inverse limit is the set:
<math display=block>\varprojlim G_i = \left\{(g_i)_{i \in I} \in {\textstyle\prod\limits_{i \in I}} G_i : f^j_i (g_j) = g_i \text{ for all } i\leq j\right\}</math>
equipped with the [[Subspace topology|relative]] [[product topology]].

One can also define the inverse limit in terms of a [[universal property]]. In [[category theory|categorical]] terms, this is a special case of a [[filtered category|cofiltered limit]] construction.

===Second definition (axiomatic)===

A profinite group is a [[compact group|compact]] and [[totally disconnected]] topological group:<ref name=":0">{{Cite web|url=https://www.math.ucdavis.edu/~osserman/classes/250C/notes/profinite.pdf|title=Inverse limits and profinite groups|last=Osserman|first=Brian|website=University of California, Davis|archive-url=https://web.archive.org/web/20181226233013/https://www.math.ucdavis.edu/~osserman/classes/250C/notes/profinite.pdf|archive-date=2018-12-26}}</ref> that is, a topological group that is also a [[Stone space]].

===Profinite completion===

Given an arbitrary group <math>G</math>, there is a related profinite group <math>\widehat{G},</math> the {{em|{{visible anchor|profinite completion}}}} of <math>G</math>.<ref name=":0" /> It is defined as the inverse limit of the groups <math>G/N</math>, where <math>N</math> runs through the [[normal subgroup]]s in <math>G</math> of finite [[Index of a subgroup|index]] (these normal subgroups are [[partial order|partially ordered]] by inclusion, which translates into an inverse system of natural homomorphisms between the quotients).

There is a natural homomorphism <math>\eta : G \to \widehat{G}</math>, and the image of <math>G</math> under this homomorphism is [[dense set|dense]] in <math>\widehat{G}</math>. The homomorphism <math>\eta</math> is injective if and only if the group <math>G</math> is [[residually finite group|residually finite]] (i.e.,
<math>\bigcap N = 1</math>, where the intersection runs through all normal subgroups <math>N</math> of finite index).

The homomorphism <math>\eta</math> is characterized by the following [[universal property]]: given any profinite group <math>H</math> and any continuous group homomorphism <math>f : G \rightarrow H</math> where <math>G</math> is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique [[continuous function (topology)|continuous]] group homomorphism <math>g : \widehat{G} \rightarrow H</math> with <math>f = g \eta</math>.

===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>

===Surjective systems{{anchor|Surjective system}}===

In practice, the inverse system of finite groups is almost always {{em|{{visible anchor|surjective inverse system|text=surjective}}}}, meaning that all its maps are surjective. Without loss of generality, it suffices to consider only surjective systems since given any inverse system, it is possible to first construct its profinite group <math>G,</math> and then {{em|reconstruct}} it as its own profinite completion.