There does not exist a nilpotent group generated by two elements such that every nilpotent group generated by two elements is a quotient

Prove that there cannot be a nilpotent group N generated by two elements such that every nilpotent group which is generated by two elements is a quotient of N. (That is, the specification of c in the definition of free nilpotent groups is necessary.)


Suppose N is such a group. Recall that D_{2 \cdot 2^k} is generated by two elements and has nilpotence class k. By our hypothesis, there is a surjective group homomorphism \varphi : N \rightarrow D_{2 \cdot 2^k}. Now \varphi[N^t] = \varphi[N]^t = D_{2 \cdot 2^k}^t for all t. Suppose N has nilpotence class c; then 1 = \varphi[1] = \varphi[N^c] = D_{2 \cdot 2^{c+1}}^c \neq 1, a contradiction. Thus no such N exists.

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