Editing Profinite group (section)

{{Short description|Topological group that is in a certain sense assembled from a system of finite groups}}
In [[mathematics]], a '''profinite group''' is a [[topological group]] that is in a certain sense assembled from a system of [[finite group]]s.

The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists <math>d\in\N</math> such that every group in the system can be [[generating set of a group|generated]] by <math>d</math> elements.<ref>{{cite arXiv |last=Segal |first=Dan |date=2007-03-29 |title=Some aspects of profinite group theory |eprint=math/0703885 }}</ref> Many theorems about finite groups can be readily generalised to profinite groups; examples are  [[Lagrange's theorem (group theory)|Lagrange's theorem]] and the [[Sylow theorems]].<ref>{{Cite book |last=Wilson |first=John Stuart |title=Profinite groups |date=1998 |publisher=Clarendon Press |isbn=9780198500827 |location=Oxford |oclc=40658188}}</ref>

To construct a profinite group one needs a system of finite groups and [[group homomorphism]]s between them. Without loss of generality, these homomorphisms can be assumed to be [[Surjective function|surjective]], in which case the finite groups will appear as [[quotient group]]s of the resulting profinite group; in a sense, these quotients approximate the profinite group.

Important examples of profinite groups are the [[abelian group|additive group]]s of [[P-adic number|<math>p</math>-adic integers]] and the [[Galois group]]s of infinite-degree [[field extension]]s.

Every profinite group is [[Compact space|compact]] and [[Totally disconnected space|totally disconnected]]. A non-compact generalization of the concept is that of [[locally profinite group]]s. Even more general are the [[totally disconnected group]]s.