Exercise 17, § 9

Let G be a Lie group. We are going to show that G has the following property:

(T) For every completely regular topological space E on which G operates properly, and every point $x$ in E, there exists a transversal at $x$ (Exerc. 16). a) Show that every Lie group having a faithful finite dimensional linear representation satisfies (T) (use Exerc. 15 and $16b$), as well as Prop. 6).

b) Prove that, if G has a closed normal subgroup N such that $G/N$ satisfies (T) and such that KN satisfies (T) for every compact subgroup K of G, then G satisfies (T) (apply Exerc. $16c$)$)$.

c) If $G_0$ is compact, then G satisfies (T).

d) Show that, if G has a discrete normal subgroup N such that $G/N$ satisfies (T), then G satisfies (T).

e) Show that G satisfies (T) (let N be the kernel of the adjoint representation; prove that $G/N_0$ satisfies (T), then apply $a$)$,b)$ and Exerc. 9 of §1).