= Adjoining a root by an HNN extension
{title2=$h^n=g$}
To adjoin an $n$th root of $g\in G$, let $m$ be the <order of a group element> $g$ and take a <cyclic group> $C=\langle a\rangle$ of order $nm$ when $m$ is finite, or infinite order when $g$ has infinite order. In the <free product> $G*C$, $\langle a^n\rangle$ and $\langle g\rangle$ have the same order. The <HNN extension> with $t^{-1}a^nt=g$ therefore embeds its base by <Britton's lemma>, and $h=t^{-1}at$ satisfies $h^n=g$. If $g=1$, take $C$ of order $n$ and identify the two trivial subgroups. This is an existence construction, not an algorithm for finding an unknown element order.
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