Conjugation preserves element order
= Conjugation preserves element order
{title2=$\operatorname{ord}(hah^{-1})=\operatorname{ord}(a)$}
For every positive integer $n$, cancellation gives $(hah^{-1})^n=ha^nh^{-1}$. Thus the power on one side is the <identity element> exactly when $a^n=e$, proving preservation of the <order of a group element>, including infinite order. In particular, $ab$ and $ba$ are conjugate because $a^{-1}(ab)a=ba$, and hence have the same order.