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Math Help - Abstract Help

  1. #1
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    Abstract Help

    1. Let σ be the 12-cycle (1 2 3 4 5 6 7 8 9 10 11 12). For which positive integers i is σi also a 12-cycle?

    2.
    If τ = (1 2)(3 4 5) determine whether there is a n-cycle σ (n ≥ 5) with τ = σ^k for
    some integer k.


    Please help.
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  2. #2
    MHF Contributor

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    I consider the first question. More generally, when is the k-th power of an n-cycle \sigma also an n-cycle ?

    We may assume that \sigma=(1 2 \cdots n).

    Consider the "orbit" \mathcal{O}=\{1,\sigma^k(1),\sigma^{2k}(1),\ldots\  } (these are the images of 1 when \sigma^k is repeatedly applied; it may help to draw a sketch in some case, like n=8, k=2 and then k=3).

    Saying that \sigma is an n-cycle amounts to saying that |\mathcal{O}|=n (this is the cardinality of \mathcal{O}).

    The cardinality of \mathcal{O} is given by the first m\geq 1 such that \sigma^{km}(1)=1, which means that n divides km. Indeed (in case it is not clear), applying \sigma is like adding 1 modulo n; then applying \sigma^{km} is like adding km modulo n, and \sigma^{km}(1)=1 iff 1+km = 1 modulo n. We have thus: |\mathcal{O}|=\min\{m\geq 1|n\mbox{ divides }km\}.
    Now one can easily check that this quantity is equal to n if, and only if n and k are relatively prime. (Or, more precisely |\mathcal{O}|=\frac{n}{\gcd(n,k)}).

    As a conclusion, the k-th power of an n-cycle is an n-cycle if, and only if n and k are relatively prime.


    It may not be the expected method (if n=12, you may just look at every case individually), but I found it interesting to look at.
    Laurent.
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