Exercise 8, § 3

Let G be a separated topological group, H a closed subgroup of G contained in the center and such that there exists a continuous surjective homomorphism $\varphi : \mathbf{R} \to G/H$. Prove that G is commutative (if $a_0$ is an element of G not in H, observe that there exists a sequence of elements $(a_n)$ of G and a sequence of elements $(c_n)$ of H such that $a_n = c_n a_{n+1}^2$, and that the subgroup of G generated by H and the $a_n$ is dense in G).