Thread: Proof regarding cycles of even or odd length

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

what you want to prove, is that if α is a n-cycle, with α = (a₁ a₂ ... a_n),

then α^k sends a_j to a_{j+k (mod n)} (you can do this by induction on k).

now if n is even, say n = 2m, then α² sends:

a₁→a₃→...→a_n-1→a₁ (since n-1 = 2m -1 is odd)

a₂→a₄→...→a_n→a₂

that is α² = (a₁ a₃ ... a_n-1)(a₂ a₄ ... a_n),

so α splits into 2 disjoint m-cycles.

but if n is odd, say n = 2m+1, so that n-2 is odd, then:

α² = (a₁ a₃ .... a_n-2 a_n a₂ a₄ ... a_n-1),

which is an n-cycle.

Thank you, that's very helpful.