I've seen a statement that if $G/C(G)$ is cyclic, then $G$ is abelian, and I could prove this. And then I was confused because if G is abelian, then it should be the same with its center $C(G)$. So the quotient group just becomes a trivial group.
Is there an abelian group whose center is not the same with itself? Orelse the statement means nothing i guess.