---
tag: CG9EH0TN27
kind: concept
name: Profinite completion
msc: ["20E18", "20E26"]
sources:
  - url: https://en.wikipedia.org/wiki/Profinite_integer
---

## Summary

The profinite completion of a [[group|G5S1M9]] $G$ is the inverse limit of all its finite quotient groups.
It is a [[profinite group|KA0YDB6QXE]] that records every finite quotient of $G$ at once, together with a natural homomorphism from $G$.
This homomorphism is injective exactly when $G$ is [[residually finite|HEE922EHQ8]].
The profinite completion of $\Z$ is the group $\hat\Z$ of profinite integers, which is the product of the $p$-adic integers over all primes $p$.

## Explanations

### L1 human

A group can be studied through its finite images, such as the integers modulo $n$ for the group of integers.
The profinite completion collects all of these finite pictures into one object.
An element of it is a compatible choice of an image in every finite quotient.
For the integers, it is a choice of a residue modulo every $n$, consistent under reduction, and such choices form the group $\hat\Z$.