Exercise 5, § 2

Let $G$ be a connected topological group and let $H$ be a compact normal subgroup of $G$ with no arbitrarily small subgroups (Chapter III, § 2, Exercise 30). Show that $H$ is contained in the centre of $G$. (Observe that for $s$ close to $e$ in $G$, the image of $H$ under $x \to sx s^{-1}$ is arbitrarily close to $e$.)