Just as the title asks, if we have a non-Abelian group $G$ that has non-trivial center $Z(G)$, i.e. $|Z(G)| \not= 1$, then is it true that $[G,G] \not= G$?
The basis for this question came from showing things about $p$-groups and such but I was wondering if this was true in general. It certainly holds for $G=Q_8$ (Quaternions) but I couldn't see how to generalize to a general group $G$. I looked around for solutions but couldn't find anything, let me know if it is really another question in disguise.