Exercise 8, § 7

Let $X$ be a set; for every integer $q \geqslant 0$ let $\mathrm{Mo}(X)_q$ be the set of elements of $\mathrm{Mo}(X)$ of length $q$ and let $\mathrm{Mo}^{(q)}(X)$ be the union of $\mathrm{Mo}(X)_{nq}, n \in \mathbf{N}$. The injection $\mathrm{Mo}(X)_q \to \mathrm{Mo}^{(q)}(X)$ extends to a homomorphism

show that this homomorphism is bijective if $q \geqslant 1$.