First the problem:
If I am correct the problem is simply asking about a specific example, one that does not necessarily generalize. I can easily show that, for a in H, aH = Ha = H for any subgroup, due to closure of H. And, as it happens I have showed (by construction) that for any element a of that is not in H, . (I note that this property is specific to the subgroup H. If I take a new cyclic subgroup K = <(123)> of that aK = Ka for all a in .) (Note: please forgive me if my cyclic notation is off. I rarely use it.)Let H be the cyclic subgroup (of order 2) of generated by . Then no left coset of H (except H itself) is also a right coset. There exist an element a in such that .
The last part is bugging me. is a true statement for all a in ! So why are they commenting on it??