Factor group of a center of a abelian group is cyclic.

I am trying understand this proof http://www.proofwiki.org/wiki/Quotient_of_Group_by_Center_Cyclic_implies_Abelian

but I am confused what its trying to prove.

Wouldn't $G/Z(G)$ group have just one element $G$ if $G$ is abelian because $Z(G)$ is all of $G$ for abelian groups. So is $Z/Z(G)$ a cyclic group with just one element? I am not quite sure what that theorem is saying.

Solution 1:

The theorem is saying that if $G/Z(G)$ is cyclic then $G$ is abelian so $G/Z(G)$ has one element. Putting these together: if $G/Z(G)$ is cyclic then it has one element.

The hypothesis here is not that $G$ is abelian. That is a consequence of the assertion that $G/Z(G)$ is cyclic. The interesting result here is that $G/Z(G)$ can never be a nontrivial cyclic group, like $\mathbb{Z}/12\mathbb{Z}$ or $\mathbb{Z}/2013\mathbb{Z}$.

The title of the theorem is not "factor group of an abelian group by its center is cyclic". It is "if the factor by the center is cyclic, then the group was abelian (so the factor was trivial)."