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

  $$\underset{⟵}{lim}{H}_i:=\left\{(h_i)\in \prod _{i\in I}{H}_i\right|{\alpha }_{ij}(h_j)=h_i\text{ for all }i\le j\}$$

  under componentwise multiplication.

• The isomorphism from $G$ to $\underset{⟵}{lim}{H}_i$ (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.