Conjugation preserves element order ($\mathrm{ord}(ha{h}^{-1})=\mathrm{ord}(a)$)

For every positive integer $n$, cancellation gives $(ha{h}^{-1}{)}^n=h{a}^n{h}^{-1}$. 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.