Exercise 8, § 4

Let $G$ be a finite-dimensional Lie group. Suppose that $L(G)$ is simple. Let $A$ be a normal subgroup of $G$. If $A$ is not open, $A$ is discrete and its centralizer in $G$ is open. (Let $a$ be the subalgebra tangent to $A$ at $e$. Show that $a = \{0\}$. Let $x$ be an element of $A$ not belonging to the centre of $G$. Consider the mapping $y \mapsto yxy^{-1}$ of $G$ into $A$. Deduce from the relation $a = \{0\}$ that the centralizer of $x$ in $G$ is open. Then, using an exponential mapping, show that the centralizer of $x$ in $G$ contains a neighbourhood of $e$ independent of $x$.