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Math Help - permutation

  1. #1
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    permutation

    How to prove the following theorem. It is based on permutation (advanced abstract algebra)

    Let n>=2 and sigma belongs to Sn be a cycle. Show that sigma is a K-cycle if and only if order of sigma is K.

    Please give a proof.
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  2. #2
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    Re: permutation

    suppose that |σ| = k, and that σ = (a1 a2....am)

    then σn sends aj to aj+n (mod m)

    since σk = id, we have:

    aj = aj+k (mod m), for all j. in particular, 1 = 1+k (mod m) which means that k is a multiple of m.

    since k is the least positive power of σ that equals the identity, 1+k is the smallest number > 1 such that 1 = 1+k (mod m), which shows that k = m, so σ is a k-cycle.

    if, on the other hand, σ is a k-cycle, then for n < k:

    σn sends a1 to a1+n (mod k), and since n < k, n+1 ≤ k, so σn(a1) = an+1 ≠ a1, so σn is not the identity.

    however, σk(aj) = aj+k (mod k) = aj, so σk = id, thus |σ| = k.

    (we need k ≥ 2 so that σ1 is not a 1-cycle, since 1-cycles ARE the identity).
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