Proposition 3

For all $w \neq 1$ in $F(X)$, there exist a finite $p$-group $G$ and a homomorphism $f$ of $F(X)$ into $G$ such that $f(w) \neq 1$.

There exist elements $x_1, \ldots, x_r$ of $X$ and integers $n_1, \ldots, n_r$ such that $w = x_1^{n_1} \ldots x_r^{n_r}$. Let $Y = \{x_1, \ldots, x_r\}$. The canonical injection of $Y$ into $X$ extends to a homomorphism $\alpha : F(Y) \to F(X)$; on the other hand, let $\beta$ be the homomorphism of $F(X)$ into $F(Y)$ whose restriction to $Y$ is the identity and which maps $X - Y$ to $\{1\}$. Then $\beta(\alpha(y)) = y$ for $y \in Y$ and hence $\beta \circ \alpha$ is the identity automorphism of $F(Y)$. There obviously exists $w'$ in $F(Y)$ such that $w = \alpha(w')$; then $\beta(w) = w' \neq 1$; now $\bigcap_{n \geqslant 1} F_n^{(p)}(Y) = \{1\}$ and there therefore exists an integer $n \geqslant 1$ such that $\beta(w) \notin F_n^{(p)}(Y)$. By Proposition 2, the group $G = F(Y)/F_n^{(p)}(Y)$ is a finite $p$-group. If $f$ is the composition of $\beta$ and the canonical homomorphism of $F(Y)$ onto $G$, then $f(w) \neq 1$.