Is there a name for groups $G$ which satisfy the property that for any $a$ and $b$ in $G$, there is a $c\in G$ and integers $n$ and $m$ such that $a=c^n$ and $b=c^m$? Such a group has to be abelian, and it's not hard to prove that this is equivalent to the property that every finitely generated subgroup is cyclic.

As a generalization, has anyone studied groups $G$ in which for any three elements $a,b,c\in G$, there are two other elements $x$ and $y$ such that $a$, $b$ and $c$ all lie in the subgroup generated by $x$ and $y$?