Group whose derived subgroup is contained in the square of its center

Definition
A group $$G$$ is termed group whose derived subgroup is contained in the square of its center if it satisfies the following equivalent conditions:


 * 1) For every $$x,y \in G$$, there exists $$z$$ in the center $$Z(G)$$ such that $$z^2 = [x,y]$$, where $$[x,y]$$ denotes the commutator of $$x$$ and $$y$$.
 * 2) The derived subgroup $$G'$$ is contained in the subgroup $$\{z^2 \mid z \in Z(G) \}$$ where $$Z(G)$$ is the center of $$G$$.