I'm having difficulty expressing the proof, even though the statement to be proved seems obvious to me.

Prove: is a cycle iff the length of the cycle is odd.

What I've got in my head is that if then . Then if we add another 2 elements c and d, , making . The pattern will obviously continue, giving 2 disjoint cycles where the elements alternate for a beginning cycle of any even length. I just need some help assembling my rambling intuitive understanding of the problem into a proper proof.

EDIT: fixed tex tags