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  "body": "\n## Statement\n\nLet $p$ be a prime.\nFor a [[pro-p group|C13D6W3MM9]] $U$ and $k \\ge 1$, let $\\overline{U^k}$ be the closed subgroup generated by all $k$-th powers.\nA 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 $p = 2$.\nIt is *uniform* if it is finitely generated, powerful and torsion-free.\nLet $G$ be a topological group.\nThen $G$ is a [[p-adic Lie group|3AKM6EPV2F]] if and only if $G$ has an [[open subgroup|QYCFH68RD1]] that is a uniform pro-$p$ group.\n\n## Remarks\n\nIn one direction, a $p$-adic Lie group has small open subgroups on which the logarithm and exponential maps converge.\nA small enough such subgroup is uniform.\nIn the other direction, fix a minimal topological generating set $u_1, \\dots, u_d$ of a uniform group $U$.\nEvery element of $U$ is a unique product $u_1^{\\lambda_1} \\cdots u_d^{\\lambda_d}$ with $\\lambda_i \\in \\Z_p$.\nIn these coordinates the group law is analytic.\nExample: 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.\nThe result was proved by Michel Lazard in his work on $p$-adic analytic groups.\nFor a [[profinite group|KA0YDB6QXE]], it turns a geometric condition, being a $p$-adic manifold, into a purely group-theoretic one.\n\n## Explanations\n\n### L1 human\n\nA $p$-adic Lie group is a group built from the $p$-adic numbers in a smooth way.\nLazard's theorem says that one can recognize these groups by algebra alone.\nA group is of this kind exactly when it has an open subgroup of a special algebraic type, called a uniform pro-$p$ group.\nSuch a group is a limit of finite groups of $p$-power order.\n\n### L2 human\n\nThe 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.\nAmong compact groups, it shows that the compact $p$-adic Lie groups are the profinite groups that contain an open uniform pro-$p$ subgroup.\nThe dimension of $G$ as a manifold equals the minimal number of topological generators of any uniform open subgroup.\nThe Lie algebra of $G$ can be built from a uniform subgroup $U$ by turning $U$ into a $\\Z_p$-lattice with a Lie bracket.\nThis is a close analogue of the real Lie group–Lie algebra correspondence.\n",
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