---
tag: DSK9GN63GQ
kind: statement
role: theorem
name: Lazard's theorem on p-adic analytic groups
msc: ["22E20", "20E18"]
uses: [3AKM6EPV2F, C13D6W3MM9, QYCFH68RD1, KA0YDB6QXE]
about: [3AKM6EPV2F, C13D6W3MM9]
sources:
  - url: https://en.wikipedia.org/wiki/Pro-p_group
---

## Statement

Let $p$ be a prime.
For a [[pro-p group|C13D6W3MM9]] $U$ and $k \ge 1$, let $\overline{U^k}$ be the closed subgroup generated by all $k$-th powers.
A finitely generated pro-$p$ group $U$ is *powerful* if $U / \overline{U^p}$ is abelian for odd $p$, or $U / \overline{U^4}$ is abelian for $p = 2$.
It is *uniform* if it is finitely generated, powerful and torsion-free.
Let $G$ be a topological group.
Then $G$ is a [[p-adic Lie group|3AKM6EPV2F]] if and only if $G$ has an [[open subgroup|QYCFH68RD1]] that is a uniform pro-$p$ group.

## Remarks

In one direction, a $p$-adic Lie group has small open subgroups on which the logarithm and exponential maps converge.
A small enough such subgroup is uniform.
In the other direction, fix a minimal topological generating set $u_1, \dots, u_d$ of a uniform group $U$.
Every element of $U$ is a unique product $u_1^{\lambda_1} \cdots u_d^{\lambda_d}$ with $\lambda_i \in \Z_p$.
In these coordinates the group law is analytic.
Example: for an odd prime $p$, the congruence subgroup $1 + p M_n(\Z_p)$ of $GL_n(\Z_p)$ is open and uniform, which shows again that $GL_n(\Z_p)$ is a $p$-adic Lie group.
The result was proved by Michel Lazard in his work on $p$-adic analytic groups.
For a [[profinite group|KA0YDB6QXE]], it turns a geometric condition, being a $p$-adic manifold, into a purely group-theoretic one.

## Explanations

### L1 human

A $p$-adic Lie group is a group built from the $p$-adic numbers in a smooth way.
Lazard's theorem says that one can recognize these groups by algebra alone.
A group is of this kind exactly when it has an open subgroup of a special algebraic type, called a uniform pro-$p$ group.
Such a group is a limit of finite groups of $p$-power order.

### L2 human

The theorem is the basis of the group-theoretic treatment of $p$-adic analytic groups, as in the book of Dixon, du Sautoy, Mann and Segal.
Among compact groups, it shows that the compact $p$-adic Lie groups are the profinite groups that contain an open uniform pro-$p$ subgroup.
The dimension of $G$ as a manifold equals the minimal number of topological generators of any uniform open subgroup.
The Lie algebra of $G$ can be built from a uniform subgroup $U$ by turning $U$ into a $\Z_p$-lattice with a Lie bracket.
This is a close analogue of the real Lie group–Lie algebra correspondence.