---
tag: C13D6W3MM9
kind: concept
name: Pro-p group
msc: ["20E18", 22C05]
specializes: [KA0YDB6QXE]
sources:
  - url: https://en.wikipedia.org/wiki/Pro-p_group
same_as:
  wikidata: Q7246615
  wikipedia: Pro-p group
---

## Summary

For a prime $p$, a pro-$p$ group is a [[profinite group|KA0YDB6QXE]] that is an inverse limit of finite [[p-groups|06QYRTX9KQ]].
The $p$-adic integers are the basic example.
Pro-$p$ groups occur as Sylow subgroups of profinite groups and as Galois groups of maximal $p$-extensions of fields.
Every $p$-adic analytic group has an open subgroup that is a pro-$p$ group.
Much of the theory of finite $p$-groups carries over to them.

## Explanations

### L1 human

Fix a prime number $p$, such as $2$ or $3$.
A finite $p$-group is a finite group whose number of elements is a power of $p$.
A pro-$p$ group is built from a compatible tower of such finite groups, much as the $p$-adic integers are built from the integers modulo $p$, $p^2$, $p^3$, and so on.
Each finite quotient of a pro-$p$ group by an open normal subgroup has order a power of $p$.