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Thread: Order of a m-cycle

  1. #1
    Super Member Bernhard's Avatar
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    Order of a m-cycle

    Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

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


    Can anyone help with this problem? Would be grateful for help!

    I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

    Peter
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Re: Order of a m-cycle

    Quote Originally Posted by Bernhard View Post
    Dummit and Foote Section 1.3 Symmetric Groups - Exercise 10 states:

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


    Can anyone help with this problem? Would be grateful for help!

    I think I can see how this works but I am struggling with how to write a clear and valid formal proof.

    Peter
    Note that $\displaystyle \sigma^m=e$ and $\displaystyle \sigma^n\neq e$ for $\displaystyle n$ ranging from $\displaystyle 1$ to $\displaystyle m-1$.
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Order of a m-cycle

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    Yes, can see that! Would that constitute a satisfactory proof?

    Peter
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  4. #4
    MHF Contributor alexmahone's Avatar
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    Re: Order of a m-cycle

    Quote Originally Posted by Bernhard View Post
    Would that constitute a satisfactory proof?
    Depends on whether you have a satisfactory argument for $\displaystyle \sigma^n\neq e$ for $\displaystyle n$ ranging from $\displaystyle 1$ to $\displaystyle m-1$.
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  5. #5
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    Re: Order of a m-cycle

    if you prove the first part, you'll have a satisfactory proof of the second part.
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