## 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 $S$.