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Thread: m cycle

  1. #1
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    m cycle

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

    I am not sure about this. If we take $\displaystyle a_k$ 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

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

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

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