profinite group

1 Definition

A topological groupMathworldPlanetmath G is profinite if it is isomorphicPlanetmathPlanetmathPlanetmath to the inverse limitMathworldPlanetmathPlanetmath of some projective system of finite groupsMathworldPlanetmath. In other words, G is profinite if there exists a directed set I, a collectionMathworldPlanetmath of finite groups {Hi}iI, and homomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath αij:HjHi for each pair i,jI with ij, satisfying

  1. 1.

    αii=1 for all iI,

  2. 2.

    αijαjk=αik for all i,j,kI with ijk,

with the property that:

The topology on a profinite group is called the profinite topology.

2 Properties

One can show that a topological group is profinite if and only if it is compactPlanetmathPlanetmath and totally disconnected. Moreover, every profinite group is residually finite.

Title profinite group
Canonical name ProfiniteGroup
Date of creation 2013-03-22 12:48:50
Last modified on 2013-03-22 12:48:50
Owner djao (24)
Last modified by djao (24)
Numerical id 9
Author djao (24)
Entry type Definition
Classification msc 20E18
Classification msc 22C05
Synonym profinite
Related topic InverseLimit
Defines profinite topology