Exercise 21, § 9

Let G be a compact Lie group, T a maximal torus of G.

a) Let $\mathscr{S}$ be a set of closed subgroups of G, stable under conjugation, such that the family of subgroups $(S\cap T)_{S\in\mathscr{S}}$ is finite. Show that the set of conjugacy classes of the subgroups $S_0$, for $S\in \mathscr{S}$, is finite (by using Exerc. $12a$)$)$, reduce to the case in which the subgroups $S_0$ are normal; consider the groups $C(S_0)_0$ and $D(S_0)$, and apply Exerc. $12b))$.

b) Show that $\mathscr{S}$ is the union of a finite number of conjugacy classes of subgroups of G (by using $a$), reduce to the case in which the subgroups $S\in \mathscr{S}$ all have the same identity component $\Sigma$, which is normal in G; then bound the orders of the elements of the groups $S/\Sigma$, and apply Exerc. $20b))$.

c) Let E be a separated topological space on which G operates continuously. Show that if the elements of E have only a finite number of orbit types for the operation of T, the same is true for the operation of G.