Does the order of the subgroup generated by two elements divide the product of the element orders? Let $(G, \cdot)$ be a group, $a,b\in G$ such that $\DeclareMathOperator{\ord}{ord}\ord(a),\ord(b)<\infty$.
Do we then have that $\left|\left<a,b\right>\right|$ divides $\ord(a)\ord(b)$? (Where $\left< a,b \right>$ denotes the subgroup generated by $a$ and $b$)
I managed to show this for the case that $a$ and $b$ commute, however I would be interested if this holds even if they don't.

$S_3=\langle (12), (23)\rangle$.
