Theorem
Let $G$ be a group such that $G/G'$ is a divisible group of finite "general" rank. Suppose also that $G''={1}$. Then $G'\leq Z(G)$.
Is it possible? How can we show that? (I really have no ideas.)
Edit
Mal'cev (Mal'cev, "On groups of finite rank" Math. Sb. 22, 351-352 (1948)) defines the "general rank" of a group $G$ to be either $\infty$ or the least positive integer $R$ such that every finitely generated subgroup is contained in a $R$-generated subgroup of $G$.