Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 1 / 4 / a / Solution

Let $m$ be the order of a group element $g$, allowing $m=\infty$. For finite $m$, take $C=⟨a\mid a^{nm}=1⟩$; for infinite $m$, take $C=⟨a\mid ⟩\cong \mathbb{Z}$. SetThe normal form theorem for a free product embeds both factors, so $G$ is a free factor. The cyclic subgroup $A=⟨a^n⟩$ has order $m$, just as $D=⟨g⟩$ does. Hence $a^{nr}\mapsto g^r$ is a well-defined group isomorphism $A\to D$, and we may form the HNN extensionThe single displayed conjugacy relation implements the full cyclic isomorphism by taking integer powers. Britton's lemma embeds $G_1$ in $H$, so the original $G$ is preserved. Now takeCancellation of successive $t{t}^{-1}$ gives $h^n=t^{-1}{a}^{n}t=g$. This proves the adjoining a root by an HNN extension construction and all required embedding claims. If $g=1$, then $m=1$, $C=C_n$, and $A=D=\{1\}$; the HNN relation is vacuous and the same calculation gives $h^n=1$. The construction also works for $n=1$. It uses the actual order of $g$ as an existence parameter, not an algorithm for determining an unknown order.