A confusion on a Theorem

Question

Theorem: Prove that if $G/Z(G)$ is cyclic then $G$ is abelian.

My confusion is that if theorem is true i.e. $G/Z(G)$ is cyclic then $G$ is abelian, then $Z(G)=G$ implies $|G/Z(G)|$ always equal to 1. Am I thinking correctly?

If I am then let $|G|=pq$ where $p,q$ are distinct primes. Then if $|Z(G)|=p$ then $|G/Z(G)|=q$ implies $|G/Z(G)|$ is cyclic. Hence $G$ is abelian. Is this true?

If this is true then $|Z(G)|=pq$. What is going on?

Answer 1

Yes. In other words, the factor group $G / Z(G)$ cannot be simultaneously cyclic and nontrivial.

Answer 2

The theorem in the question is usually stated in the same form, but it is better to say it as

If $G$ is non-abelian then $G/Z(G)$ should not be cyclic.

With this statement, you may immediately solve your question. Let $|G|=pq$. If $|Z(G)|=p$, then $Z(G)<G$, i.e. $G$ should be non-abelian. Then $|G/Z(G)|=q$ i.e. $G/Z(G)$ is cyclic, contradiction.