Universal property of free nilpotent groups of finite nilpotence class

Let S be a set and c a positive integer. Formulate the notion of a free nilpotent group on S of nilpotence class c and prove it has the appropriate universal property with respect to nilpotent groups of class at most c.


Define the group N(S) = (S,R), where R = \{ F(S)^c \}. (Exponents denoting higher commutator subgroups.)

We will now prove the following: If G is a nilpotent group of class at most c and \varphi : S \rightarrow G is a set function, then there exists a unique group homomorphism \Phi : N(S) \rightarrow G extending \varphi.

To see this, recall from the universal property of free groups that there is a unique group homomorphism \Psi : F(S) \rightarrow G extending \varphi. Moreover, \Psi[F(S)^c] \leq G^c = 1, since G is nilpotent of class at most c. Thus there is a unique group homomorphism \Phi : N(S) \rightarrow G such that \Phi \circ \pi = \Psi; identifying s \in S with sR \in N(S), we can say that \Phi extends \varphi. This gives existence.

To see uniqueness, Note that any group homomorphism \Phi : N(S) \rightarrow G which extends \varphi is determined uniquely by \Psi, which is determined uniquely by S.

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