# Equivalence of definitions of profinite group

This article gives a proof/explanation of the equivalence of multiple definitions for the term profinite group
View a complete list of pages giving proofs of equivalence of definitions

## Statement

The following are equivalent for a topological group:

1. It is the inverse limit of an inverse system of finite groups, each equipped with the discrete topology.
2. It is a compact totally disconnected T0 topological group.

A topological group satisfying both equivalent conditions is termed a profinite group.

## Proof

### (1) implies (2)

Given: An inverse system $G_i,i \in I$ of finite groups with discrete topologies. $G$ is the inverse limit of the system.

To prove: $G$ is compact, $T_0$, and totally disconnected.

Proof

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $G$ can be identified with a closed subgroup of the product $\prod_{i \in I} G_i$, equipped with the product topology and the direct product group structure. definition of inverse limit $G$ is an inverse limit of the $G_i$s
2 $G$ is Hausdorff (hence $T_0$) Facts (1), (2) all the $G_i$s are finite and discrete, hence Hausdorff Step (1) By Fact (1), the product $\prod_{i \in I} G_i$ is Hausdorff, hence by Fact (2), so is any subgroup of that. By Step (1), we thus get that $G$ is Hausdorff.
3 $G$ is compact Facts (3), (4) all the $G_i$s are finite, hence compact Step (1) By Fact (3), the product $\prod_{i \in I} G_i$ is compact, hence by Fact (4), so is any closed subgroup of that. By Step (1), we thus get that $G$ is compact.
4 $G$ is totally disconnected Fact (5) all the $G_i$s are discrete, hence totally disconnected Step (1) [SHOW MORE]

### (2) implies (1)

Given: A compact $T_0$ totally disconnected group $G$.

To prove: $G$ is the inverse limit of an inverse system of finite groups.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 The open normal subgroups of $G$ form a basis at the identity element for $G$. Fact (6) $G$ is compact and totally disconnected Follows directly from Fact (6).
2 If $N$ is an open normal subgroup of $G$, then $G/N$ is a finite group with the discrete topology. Fact (7) $G$ is compact By Fact (7), $G/N$ is finite. Further, $N$ is open, and so are all its cosets, so the coset space gets a discrete quotient topology.
3 Consider the homomorphism from $G$ to the inverse limit of the inverse system of quotients of $G$ by open normal subgroups. This is continuous and has dense image.
4 The homomorphism of Step (3) is a closed map. Fact (8)
5 The homomorphism of Step (3) is injective.
6 The homomorphism of Step (3) is an isomorphism of topological groups.