Exercise 9, § 4

Let G be a compact Lie group over K of dimension n. Suppose that the Lie algebra L(G) is simple. For all g ∈ G, every integer m ≥ 1 and every neighbourhood V of e in G, let M(g, m, V) denote the set of elements of G of the form $\prod_{i=1}^m x_i(y_i, g)x_i^{-1}$ with $x_1, \ldots, x_m, y_1, \ldots, y_m$ in V.

(a) Show that if g is an element of G whose centralizer is not open, if m is an integer ≥ n and if V is a neighbourhood of e, then M(g, m, V) is a neighbourhood of e. (There exists $a \in L(G)$ such that $(\mathrm{Ad}\,g)(a) \neq a$. Let φ be an exponential mapping of G. Let y be the image of 1 under the tangent mapping at 0 to $\lambda \mapsto (\phi(\lambda a), g)$ (where $\lambda \in K$). Then $b = (\mathrm{Ad}\,g - 1)(a) \neq 0$. There exist $x_1, \ldots, x_m$ in V such that $\{(\mathrm{Ad}\,x_1)(b), \ldots, (\mathrm{Ad}\,x_m)(b)\}$ contains a basis of L(G). Consider the mapping

of $G^{2m}$ into G. Show that the tangent mapping at $(x_1, \ldots, x_m, e, \ldots, e)$ is surjective.)

(b) Let U be a neighbourhood of e in G. Let m be an integer ≥ 1. There exists an element g of G whose centralizer is not open, such that $M(g, m, G) \subset U$. (Argue by reductio ad absurdum using the compactness of G.)

(c) Let G' be a compact Lie group over K whose Lie algebra is simple. Let $\phi : G \to G'$ be a bijective homomorphism of abstract groups. Then $\phi$ is continuous. (Apply (b) to G' and (a) to G.)