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.

$\alpha_{ii}=1$ for all $i\in I$ ,
2.

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

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.

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.