profinite group
1 Definition
A topological group^{} $G$ is profinite if it is isomorphic^{} to the inverse limit^{} of some projective system of finite groups^{}. In other words, $G$ is profinite if there exists a directed set $I$, a collection^{} of finite groups ${\{{H}_{i}\}}_{i\in I}$, and homomorphisms^{} ${\alpha}_{ij}:{H}_{j}\to {H}_{i}$ for each pair $i,j\in I$ with $i\le j$, satisfying

1.
${\alpha}_{ii}=1$ for all $i\in I$,

2.
${\alpha}_{ij}\circ {\alpha}_{jk}={\alpha}_{ik}$ for all $i,j,k\in I$ with $i\le j\le k$,
with the property that:

•
$G$ is isomorphic as a group to the projective limit
$$\underset{\u27f5}{lim}{H}_{i}:=\left\{({h}_{i})\in \prod _{i\in I}{H}_{i}\right{\alpha}_{ij}({h}_{j})={h}_{i}\text{for all}i\le j\}$$ under componentwise multiplication.

•
The isomorphism from $G$ to $\underset{\u27f5}{lim}{H}_{i}$ (considered as a subspace^{} of $\prod {H}_{i}$) is a homeomorphism of topological spaces^{}, where each ${H}_{i}$ is given the discrete topology and $\prod {H}_{i}$ is given the product topology.
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 compact^{} and totally disconnected. Moreover, every profinite group is residually finite.
Title  profinite group 

Canonical name  ProfiniteGroup 
Date of creation  20130322 12:48:50 
Last modified on  20130322 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 