Exercise 22, § 7

(a) Let $(y_n)_{n \in \mathbf{Z}}$ be a basic family of a group F. Let $\phi$ be the automorphism of F which maps $y_n$ to $y_{n+1}$; this automorphism defines as operation $\tau$ of $\mathbf{Z}$ on F; let $E = F \times_\tau \mathbf{Z}$ be the corresponding semi-direct product. Show 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. 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 $f$ and $g$ are inverses of one another.)

(b) Deduce that the normal subgroup of $F_{x, y}$ generated by $y$ is a free group with basic family $(x^n y x^{-n})_{n \in \mathbf{Z}}$. Obtain this result by applying Exercise 20 to the representative system consisting of the $x^n, n \in \mathbf{Z}$.

(c) Extend the above to free groups of arbitrary rank.