Exercise 15, § 2

Let $(x_i)_{1 \leq i \leq n}$ be a finite sequence of left cancellable elements. (a) Show that the relation $x_1x_2 \ldots x_n = e$ implies all the relations $x_{i+1} \ldots x_n x_1 x_2 \ldots x_i = e$ which are obtained from one another by "cyclic permutation" for $1 \leq i \leq n$. (b) Deduce that if the composition of the sequence $(x_i)$ is invertible each of the $x_i$ is invertible.