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Math Help - m cycle

  1. #1
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    m cycle

    Prove that if \sigma is the m cycle (a_1, a_2,\cdots , a_m), then for all i\in\{1,2,\cdots , m\}, \sigma^i(a_k)=a_{k+i} where a_{k+i} is replaced by its least positive residue mod m. Deduce that |\sigma |=m.

    I am not sure about this. If we take a_k unless i is m, we wont get back to k mod m.
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  2. #2
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    Re: m cycle

    \sigma(a_k) = a_{k+1 (mod\ m)}, by definition.

    what you are being asked to prove, is that \sigma^2 takes a_1\rightarrow a_3 \rightarrow a_5 \dots a_m \rightarrow a_2

    \sigma^3 takes a_1\rightarrow a_4 \rightarrow a_7 \dots a_m \rightarrow a_3 and so on....
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