Cyclic quotient by the center
= Cyclic quotient by the center
If $G/Z(G)$ is cyclic, then $G$ is abelian. Indeed, if $gZ(G)$ generates the quotient, every element is $g^az$ with $z\in Z(G)$, and any two such elements commute.
= Cyclic quotient by the center
If $G/Z(G)$ is cyclic, then $G$ is abelian. Indeed, if $gZ(G)$ generates the quotient, every element is $g^az$ with $z\in Z(G)$, and any two such elements commute.