TITLE: If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?
QUESTION [11 upvotes]: The theorem "if $G/Z(G)$ is cyclic then $G$ is abelian" is a popular exercise.
But what is the point of this theorem if $G/Z(G)$ can only be cyclic if it is trivial?
Does "$G/Z(G)$ is cyclic" actually appears in other proofs or is it just a popular exercise?

REPLY [11 votes]: A nice application is in proving that the center of a  non-abelian cannot be too large:

If $G$ is a non-abelian finite group, then $|Z(G)| \leq \frac {1}{4} |G|$.

The contrapositive is

If $|Z(G)| > \frac {1}{4} |G|$, then $G$ is abelian.

Indeed, $|Z(G)| > \frac {1}{4} |G|$ implies that $G/Z(G)$ has order at most $3$, and so is cyclic.