Exercise 13, § 4

(a) Let $F$ be a free group with basic family $\{\bar{x}, \bar{y}\}$ of two elements, let $c$ be an integer $\geq 2$ and let $m$ be an integer $\geq 1$. We write $F^c = F/C^c F$ and denote by $x, y$ the images of $\bar{x}, \bar{y}$ in $F^c$. Let $F_m^c$ be the subgroup of $F^c$ generated by $\{x^m, y\}$. Show that there exists an integer $N \geq 0$ such that $z^{m^N} \in F_m^c$ for all $z \in F^c$ (use Exercise 11). (b) Let $I^c$ (resp. $I_m^c$) be the normal subgroup of $F^c$ (resp. $F_m^c$) generated by $y$. Show that, if $N$ is chosen as above and $z \in I^c$, then $z^{m^N} \in I_m^c$ (note that $F^c/I^c$ is an infinite cyclic group with generator the image of $x$ and deduce that $F_m^c/I_m^c \to F^c/I^c$ is injective and hence that $I_m^c = F_m^c \cap I^c$). In particular $xy^{m^N} x^{-1} \in I_m^c$. (c) Suppose that $G$ is nilpotent. Let $H$ be a subgroup of $G$ and let $L$ be a normal subgroup of $H$. Let $g \in G$ and let $m \geq 1$ be such that $g^m \in H$. Show that, if $N$ is sufficiently large, $g l^{m^N} g^{-1} \in L$ for all $l \in L$. (If $G$ is of class $< c$, choose $N$ as in (a) and use the homomorphism $f : F^c \to G$ such that $f(x) = g, f(y) = l$; note that $f(F_m^c) \subset H, f(I_m^c) \subset L$ and apply (b) above.)