# 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 $\alpha_{ij}\colon H_{j}\to H_{i}$ for each pair $i,j\in I$ with $i\leq j$, satisfying

1. 1.

$\alpha_{ii}=1$ for all $i\in I$,

2. 2.

$\alpha_{ij}\circ\alpha_{jk}=\alpha_{ik}$ for all $i,j,k\in I$ with $i\leq j\leq k$,

with the property that:

• $G$ is isomorphic as a group to the projective limit

 $\,\underset{\longleftarrow}{\lim}\,H_{i}:=\left\{\left.(h_{i})\in\prod_{i\in I% }H_{i}\ \right|\ \alpha_{ij}(h_{j})=h_{i}\ \text{ for all }\ i\leq j\right\}$

under componentwise multiplication.

• The isomorphism from $G$ to $\,\underset{\longleftarrow}{\lim}\,H_{i}$ (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 ProfiniteGroup 2013-03-22 12:48:50 2013-03-22 12:48:50 djao (24) djao (24) 9 djao (24) Definition msc 20E18 msc 22C05 profinite InverseLimit profinite topology