TITLE: In any Lie group with finitely many connected components, does there exist a finite subgroup which meets every component?
QUESTION [30 upvotes]: This question concerns a statement in a short paper by S. P. Wang titled “A note on free subgroups in linear groups" from 1981. The main result of this paper is the following theorem.
Theorem (Wang, 1981): For every field $k$ of characteristic 0 and subgroup $\Gamma$ of $GL(n, k)$, the group $\Gamma$ either has a nonabelian free subgroup or possesses a normal solvable subgroup of index < $\lambda(n)$.
To prove this theorem, Wang needs the following statement:
“It is easily shown that in any Lie group with finitely many connected components, there is a finite subgroup which meets every component.”
Please give me a proof of this statement.

REPLY [5 votes]: I'm not at all sure what S.P. Wang had in mind as an "easy" proof, but it's worth adding some explicit references to published work concerning the well-studied structure of the normalizer of a maximal torus in a compact Lie group.    (Some of this has also come up in an earlier MO question here.)  
First, the reference for Wang's short paper is:  "A note on free subgroups in linear groups". J. Algebra 71 (1981), no. 1, 232–234.   This builds on an influential 1972 paper by Tits in the same journal (unfortunately not freely accessible online).   
The paper "Normalisateurs de tores. I" by Tits (also in J. Algebra) is referenced in the compact Lie group setting by Dwyer and Wilkerson in their paper which can be freely downloaded here.   Their reference list also includes a helpful older paper by M. Curtis et al.