---
tag: 383CHJH46R
kind: definition
name: Profinite completion
msc: ["20E18", "20E26"]
defines: CG9EH0TN27
uses: [G5S1M9, 0CB0DX5BVV, 0DTXBQPA2S, 0KYX2N69J0, 0GBG8B49AA, KA0YDB6QXE, HEE922EHQ8]
sources:
  - url: https://en.wikipedia.org/wiki/Profinite_integer
---

## Definition

Let $G$ be a [[group|G5S1M9]], and let $\mathcal{N}$ be the set of [[normal subgroups|0CB0DX5BVV]] of finite [[index|0DTXBQPA2S]] in $G$.
For $N \in \mathcal{N}$ give the finite [[quotient group|0KYX2N69J0]] $G/N$ the discrete topology.
The *profinite completion* of $G$ is
$$\hat G = \varprojlim_{N \in \mathcal{N}} G/N = \Big\{ (g_N N)_{N \in \mathcal{N}} \in \prod_{N \in \mathcal{N}} G/N : g_M N = g_N N \text{ whenever } M \subseteq N \Big\},$$
with the group law taken coordinatewise and the topology induced by the [[product topology|0GBG8B49AA]].
The map $\iota \colon G \to \hat G$, $\iota(g) = (gN)_{N \in \mathcal{N}}$, is the canonical homomorphism.

## Remarks

$\hat G$ is a [[profinite group|KA0YDB6QXE]], and $\iota(G)$ is dense in $\hat G$.
The kernel of $\iota$ is the intersection of all $N \in \mathcal{N}$, so $\iota$ is injective if and only if $G$ is [[residually finite|HEE922EHQ8]].
Example: for $G = \Z$, the finite-index subgroups are $n\Z$ with $n \ge 1$, and $\hat\Z = \varprojlim \Z/n\Z$; by the Chinese remainder theorem it is isomorphic to $\prod_p \Z_p$.
Here $\iota$ is injective, since $\bigcap_n n\Z = \set{0}$.
Non-example of injectivity: $(\Q, +)$ has no proper subgroup of finite index, so $\hat\Q$ is trivial and $\iota$ is not injective.
Universal property: every homomorphism from $G$ to a profinite group $P$ factors uniquely through $\iota$ by a continuous homomorphism $\hat G \to P$.