Let be a group and . Prove that the map sends each left coset to a right coset (of ), and hence .
(This is a slightly different approach.)
Define by . Suppose ; then . and we have , so . By a lemma to a previous exercise, this induces a mapping given by . is clearly surjective, so is surjective. Now suppose ; then , so that in particular and hence . Thus . Hence is a bijection.