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Power of a cycle

  1. Nov 16, 2011 #1
    let π be a product of disjoint m-cycles. Prove that π is a power of a cycle?

    So this is like asking show that π = βx for some cycle β and pos. integer x. right?

    I don't know how to proceed on this except for the fact that the order of π is m.

    any hints please
  2. jcsd
  3. Nov 16, 2011 #2


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    what have you tried? remember, when you have no clue what will work, any idea at all is progress.
  4. Nov 16, 2011 #3
    man I have no idea.

    I know I can write π = (....)(....)(....)(....)(....)(....)(....)(....)(....)(....)

    probably need to consider when m is even and odd.

    I can break down π to a product of transpositions.

    But the end result is too abstract I can get my head around it.
  5. Nov 17, 2011 #4
    First, note that

    [tex]\sigma (i_1~i_2~i_3~...~i_n) \sigma^{-1}=(\sigma(i_1)~\sigma(i_2)~\sigma(i_3)~...~\sigma(i_n))[/tex]

    This allows you to bring everything back to the cycle (1 2 3 ... n).

    Now, take powers of (1 2 3 ... n) and see what types of disjoint cycle decompostions you meet.
  6. Nov 22, 2011 #5
    let ∏= ∏αi for all 1≤i≤n with αi = (ai1 ai2 ai3 ...aim)

    consider θ = (a11 a21......an1 a12 a22......a1m a2m a3m...anm)

    then applying θ n times will give us the original ∏.

    Hence ∏ = θn
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