No it's not right. The problem is that you are confusing the equation with the equation where c is an element of .

Looking at H rather than for a moment, the equation means that the sets aH and Ha have the same elements. This doesnotmean 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 isnottrue 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 .