Exercise 16, § 4

Let G be a locally compact connected group, let K be a compact normal subgroup of G and let N be a closed normal subgroup of K such that $K/N$ has no arbitrarily small subgroups ($\S$ 2, Exercise 30). Show that in this case $N$ is a normal subgroup of $G$. (The hypothesis on $K/N$ implies that there is a compact neighbourhood $U$ of $e$ in $G$ such that $xNx^{-1} \subset N$ for each $x \in U$).