Topological groups, Lie groups Topological groups: basic notions Theorem

Lazard's theorem on p-adic analytic groups

Builds on p-adic Lie group Pro-p group Open subgroup Profinite group

Statement

Let $p$ be a prime. For a pro-p group $U$ 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 if and only if $G$ has an open subgroup 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, it turns a geometric condition, being a $p$ -adic manifold, into a purely group-theoretic one.

Going deeper

For a student

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.

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Open subgroup concept
Pro-p group concept
Profinite group concept
p-adic Lie group concept
Lazard's theorem on p-adic analytic groups statement

Tag	`DSK9GN63GQ`
URI	`m2://DSK9GN63GQ`
Kind	statement, theorem
Writing	W0 stub
MSC	22E20, 20E18
Status	This copy of the site has no build data. Run `m2 status DSK9GN63GQ`.
About	p-adic Lie group, Pro-p group
Uses	p-adic Lie group, Pro-p group, Open subgroup, Profinite group
Changed	2 November 2026