Prove that there cannot be a nilpotent group generated by two elements such that every nilpotent group which is generated by two elements is a quotient of . (That is, the specification of in the definition of free nilpotent groups is necessary.)
Suppose is such a group. Recall that is generated by two elements and has nilpotence class . By our hypothesis, there is a surjective group homomorphism . Now for all . Suppose has nilpotence class ; then , a contradiction. Thus no such exists.