Lemma 1

Let G be a connected topological group, Z its centre and $Z'$ a discrete subgroup of Z. Then the centre of $G/Z'$ is $Z/Z'$. Let y be an element of G whose class modulo $Z'$ is a central element of

G/Z'. Let $\phi$ be the mapping $g \mapsto gyg^{-1}y^{-1}$ of G into G. Then $\phi(G)$ is connected and contained in $Z'$ and hence $\phi(G) = \phi(\{e\}) = \{e\}$. Therefore $y \in Z$.