Let $G$ be a group with identity element $e$. Assume that $G$ is non-abelian and simple, in the sense that the only normal subgroups of $G$ are $\{e\}$ and $G$. Define the centre of $G$ by $Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}$. Then $Z(G)=\{e\}$.