It is trivial that a group $G$ is abelian if and only if every subgroup of $G$ with two generators is abelian (i.e., any two elements commute).
If $G$ is a nilpotent group, every subgroup with two generators must be nilpotent. Is the reciprocal true? More precisely:
Let $G$ be a group and suppose that every subgroup of $G$ generated by two elements is nilpotent (with uniformly bounded class if needed). Is $G$ necessarily nilpotent?