---
tag: KA0YDB6QXE
kind: concept
name: Profinite group
msc: ["20E18", 22C05]
sources:
  - url: https://en.wikipedia.org/wiki/Profinite_group
same_as:
  wikidata: Q1428923
  wikipedia: Profinite group
---

## Summary

A profinite group is a topological group that is an inverse limit of finite groups.
Equivalently, it is a compact, Hausdorff and totally disconnected topological group.
Profinite groups record a whole compatible family of finite quotients at once.
The main examples are the [[p-adic integers|V6TJAFZ4MD]], the profinite completion $\hat\Z$ of $\Z$, and Galois groups of infinite Galois extensions with the Krull topology.

## Explanations

### L1 human

The integers modulo $p$, modulo $p^2$, modulo $p^3$, and so on, form a tower of finite groups, each mapping onto the previous one by reduction.
A compatible choice of one residue at each level is a $p$-adic integer, and these choices form a group.
This group is profinite: it is built entirely from finite pieces, and it carries a topology in which two elements are close when they agree at many levels.
In the same way, the Galois group of all algebraic numbers over $\Q$ is assembled from the finite Galois groups of number fields.