Exercise 22, § 7

(a) Let $(y_n)_{n \in \mathbf{Z}}$ be a basic family of a group $F$. Let $\varphi$ be the automorphism of $F$ which transforms $y_n$ into $y_{n+1}$; this automorphism defines an operation $\tau$ of $\mathbf{Z}$ on $F$; let $E = F \times_{\tau} \mathbf{Z}$ be the corresponding semidirect product. Now prove that the elements $x = (e, 1)$ and $y = (y_0, 0)$ form a basic family of $E$. (Let $F_{x, y}$ be the free group constructed on $\{x, y\}$, and let $f$ be the canonical homomorphism of $F_{x, y}$ into $E$. One will show that there exists a homomorphism $g : E \to F_{x, y}$ such that $g((0, 1)) = x$ and $g((y_n, 0)) = x^n y x^{-n}$ and that $f$ and $g$ are reciprocal to one another.) (b) Deduce that the normal subgroup of $F_{x, y}$ generated by $y$ is a free group with basic family $(x^n y x^{-n})_{n \in \mathbf{Z}}$. Recover this result by applying exerc. 20 to the system of representatives formed by the $x^n$ for $n \in \mathbf{Z}$. (c) Extend what precedes to free groups of arbitrary rank.