# Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157):

Let $G$ be a locally compact group. Then

1. (Gleason-Montgomery-Zippin-Yamabe) G is a real Lie group iff it does not contain arbitrarily small subgroups (i.e., there exists a neighbourhood of the identity containing no nontrivial subgroup).
2. (Lazard) G is a $p$-adic Lie group iff it contains an open subgroup $U$ such that $U$ is a finitely generated pro-$p$-group with $[U,U] \subset U^{p^2}$.

Are there further results that tell us when $G$ is a Lie group over $K$, $K = \mathbb{C}$ or $[K: \mathbb{Q}_p] < \infty$?

• Question: does (1) mean "$G$ is a real Lie group if and only if it does not contain subgroups of arbitrarily small but positive Haar measure"? – Mike F Jul 13 '13 at 23:39
• @Mike: I informally paraphrased the statement. Here I mean 'small' in just a general topological sense. Our statement, put more carefully, should read '$G$ is a real Lie group iff there exists a neighbourhood of the identity containing no nontrivial subgroup.' – Joshua Seaton Jul 14 '13 at 1:06
• Clarified now in an edit. – Joshua Seaton Jul 14 '13 at 1:11
• Thanks for the clarification. That's an incredible theorem! – Mike F Jul 14 '13 at 4:36
• The theorem (Gleason-Montgomery-Zippin-Yamabe) is considered to be the solution of Hilbert's Fifth Problem. – GEdgar Jul 20 '13 at 13:50

Look at the Terence Tao notes on the Hilbert fifth problem. http://terrytao.wordpress.com/books/hilberts-fifth-problem-and-related-topics/

The first chapter contains some sort of overview and explanation of philosophy behind all this.