Let

* $H$ be a subgroup of $G$;
* $a$ and $b$ belong to $G$

Then $aH = bH$ does not always imply $Ha = Hb$, as shown by a counter example of the symmetric group $3$.

I can see that this is indeed a counter example. But can anyone tell me intuitively / conceptually why it is the case? (Or am I meant to just try out a few common groups to look for contradictions, after failing to prove that it is otherwise?)

Thank you