Looking at H rather than for a moment, the equation means that the sets aH and Ha have the same elements. This does not mean that ah=ha for each element h of H. Instead, it means that, given h in H, ah=h'a for some element h' of H (which may be different from h).
Now coming back to the subgroup , it is not true that if then . What you can say is that and . You then need to show that . A similar argument will then show the reverse inequality .