Let H = {(1),(13)(24)} in A4 .

(a) Show that H is not normal in A4 .

(b) Show that (123)H = (243)H and (124)H = (132)H but that (123)(124)H 6= (243)(132)H . This proves that the group operation we defined on the set of (left) cosets G/H is not well defined unless we know that the subgroup H is normal.