Remark

Let H be a Hall set relative to X (§ 2, no. 10). Let M be the magma defined by the law of composition (x, y) \mapsto (x, y) = x^{-1}y^{-1}xy on F(X) and let $\phi$ be the homomorphism of M(X) into M whose restriction to X is the identity. The elements of $\phi(H)$ are called the basic commutators of F(X) associated with the Hall set H. For every integer n \geqslant 1, let H_n be the subset of H consisting of the elements of length n; we know (§ 2, no. 11, Theorem 1) that the canonical mapping of H_n into L_\mathbf{Z}(X) is a basis of the Abelian group L_\mathbf{Z}^n(X). Moreover, $\phi(H_n) \subset C^n$; for all m \in H_n, let $\phi_n(m)$ denote the class mod. C^{n+1} of $\phi(m) \in C^n$. Theorem 3 then shows that $\phi_n$ is a bijection of Hn onto a basis of the Abelian group* C^n/C^{n+1}. It follows immediately that, for all w \in F(X) and all i \geqslant 1, there exists a unique element $\alpha_i$ of $\mathbf{Z}^{(H_i)}$ such that, for n \geqslant 1,

where the product is calculated according to the total ordering given on H.