Proposition 2

Suppose that X is finite. For every integer n \geqslant 1, the group F(X)/Fn^{(p)}(X) is a finite p-group of nilpotency class \leqslant n.*

Arguing by induction on $n$, it suffices to prove that $F_n^{(p)}(X)/F_{n+1}^{(p)}(X)$ is a finite commutative $p$-group for all $n \geqslant 1$. For all $w \in F_n^{(p)}(X)$, the element $\gamma(w) - 1$ of $\hat{A}(X)$ is of order $\geqslant n$; we denote by $\delta_n(w)$ the homogeneous component of $\gamma(w) - 1$ of degree $n$. The mapping $\delta_n : F_n^{(p)}(X) \to A^n(X)$ is a homomorphism with kernel $F_{n+1}^{(p)}(X)$ (§ 4, no. 5, Proposition 3) and hence $F_n^{(p)}(X)/F_{n+1}^{(p)}(X)$ is isomorphic to a subgroup of $A^n(X)$. Since $X$ is finite, $A^n(X)$ is a finite-dimensional vector space over $\mathbf{F}_p$ and hence a finite commutative $p$-group and so is $F_n^{(p)}(X)/F_{n+1}^{(p)}(X)$.