Exercise 20, § 9

Let G be a compact Lie group, $m$ a positive integer.

a) Show that the set of conjugacy classes of subgroups of G of order $\leq m$ is finite (supposing this to be false, construct a finite group F and a sequence of homomorphisms $\varphi_n: F\rightarrow G$ such that $\varphi_i(F)$ is not conjugate to $\varphi_j(F)$ for $i\not=j$, and such that $\varphi_n(f)$ tends to a limit $\varphi (f)$ for all $f\in F$; show that $\varphi$ is a homomorphism, and that this leads to a contradiction with Exerc. 19). b) Show that the set of conjugacy classes of subgroups F of G all of whose elements are of order $\leq m$ is finite (let $\mu$ be a Haar measure on G, and let U be a symmetric neighbourhood of the identity element such that $U^2$ contains no non-trivial element of order $\leq m$; prove that Card(F) $\leq \mu(G)/\mu(U)$).