I just started reading about the upper and lower central series of a group. I'm wondering if there is a general relationship between the commutator subgroup $G'$ and the center $Z(G)$ of a group $G$.
There is Proposition that states that if $G'$ is not trivial and $G$ is nilpotent then $G'\cap Z(G)$ is not trivial. What if $G$ is not nilpotent?
More specifically, if the center of a group is trivial, can we deduce anything about the elements of $G'$?