Abstract Algebra | Converse of Lagrange's Theorem

[@FrankBria]

Question: If gH = Hg only for g not in H how does that mean all conjugates of elements of H are in H? Wouldn't it just show that conjugates of H are H for g not in H? So in your proof we would have to establish that the elements you're using to transform each cycle are not in H. True? Or am I missing something?

[@FrankBria]

each 3-cycle