Exercise 10, § 4

Let $x, y$ be in $G$ and let $r, s$ be two integers $\geq 1$. Suppose that $(x^r, y^s) = e$. (a) Show that, for all $n \geq 1$, $(x, y)^{(rs)^n} \in C^{n+2} G$. (It can be assumed that $G$ is generated by $\{x, y\}$; then apply Exercise 9 (b), noting that $(x^r, y^s) \equiv (x, y)^{rs}$ mod. $C^3 G$.) (b) Suppose that $x^r = y^s = e$; let $t$ denote the g.c.d. of $r$ and $s$. Show that $(x, y)^t \in C^3 G$ and deduce that $(x, y)^{tn} \in C^{n+2} G$ for all $n \geq 1$ (same method).