---
date: 2022-01-15 21:49
modification date: Wednesday 26th January 2022 23:14:05
title: profinite group
aliases: [profinite group]

---

Tags: 
#todo #todo/stub 
Refs:
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# profinite group

A **profinite group** is a topological group that is isomorphic to the inverse limit of discrete finite groups.[

Examples:

- The [p-adic integers](p-adic integers.md) 
- The [profinite integers](profinite integers.md)

Profinite completions

Given $G\in \Grp$ a related profinite group $\hat{G} \in \Grp$, the profinite completion of G. It is defined as
$$
\hat G := \inverselim_{\substack{ N\normal G \\ [G:N] < \infty }} G/N
,$$ 
where $N$ runs through the normal subgroups in G of finite index.

![](attachments/Pasted%20image%2020220126231605.png)

![](attachments/Pasted%20image%2020220417004611.png)


# Warning: on computing limits
![](attachments/Pasted%20image%2020220126231425.png)

# Examples

![](attachments/Pasted%20image%2020220126231544.png)
![](attachments/Pasted%20image%2020220126232225.png)

![](attachments/Pasted%20image%2020220408132340.png)