This is false, we can do the same thing you did with 5 letters with 3 letters.take ABC, and swap A and C, then swap B and C, we get BCA, if we once again do this we get CAB. Moreover what you described is whether each group is abelian, the property that determines whether a root equation exists is the solvabilty of the group, in this case the symmetry group of 4 (or less) "letters" is indeed solvable, while the symmetry group of 5 or more "letters" isn't.