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Math Help - Proof regarding cycles of even or odd length

  1. #1
    Junior Member beebe's Avatar
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    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: \alpha^2 is a cycle iff the length of the cycle \alpha is odd.

    What I've got in my head is that if \alpha=(a,b) then \alpha^2=(a)(b). Then if we add another 2 elements c and d, \alpha=(a,b,c,d), making \alpha^2=(a,c)(b,d). 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
    Last edited by beebe; July 20th 2012 at 08:52 AM.
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  2. #2
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    Re: Proof regarding cycles of even or odd length

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

    then αk sends aj to aj+k (mod n) (you can do this by induction on k).

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

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

    a2→a4→...→an→a2

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

    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 .... an-2 an a2 a4 ... an-1),

    which is an n-cycle.
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  3. #3
    Junior Member beebe's Avatar
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    Re: Proof regarding cycles of even or odd length

    Thank you, that's very helpful.
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