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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 $[K: \mathbb{Q}_p] < \infty$?

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  • $\begingroup$ 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"? $\endgroup$ – Mike F Jul 13 '13 at 23:39
  • $\begingroup$ @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.' $\endgroup$ – Joshua Seaton Jul 14 '13 at 1:06
  • $\begingroup$ Clarified now in an edit. $\endgroup$ – Joshua Seaton Jul 14 '13 at 1:11
  • $\begingroup$ Thanks for the clarification. That's an incredible theorem! $\endgroup$ – Mike F Jul 14 '13 at 4:36
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    $\begingroup$ The theorem (Gleason-Montgomery-Zippin-Yamabe) is considered to be the solution of Hilbert's Fifth Problem. $\endgroup$ – GEdgar Jul 20 '13 at 13:50
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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.

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