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Math Help - cycle

  1. #1
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    Post cycle

    I have to show that if sigma is a cycle of odd length then sigma^2 is a cycle.

    So this is what I assume, (and please help me check if it is an suitable answer);

    wlog, assume that
    sigma=(1,2,3,......,m)  where m is odd.
    Because m is odd, I can compute that
    sigma^2=(1,2,3,....,m)(1,2,3,....,m)(1,3,5,....,m,  2,4,6,...m-1)
    which is again a cycle.
    Is this a suitable proof, and what else can I add to it.


    Thank You
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  2. #2
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    Quote Originally Posted by Sally_Math View Post
    I have to show that if sigma is a cycle of odd length then sigma^2 is a cycle.

    So this is what I assume, (and please help me check if it is an suitable answer);

    wlog, assume that
    sigma=(1,2,3,......,m)  where m is odd.
    Because m is odd, I can compute that
    sigma^2=(1,2,3,....,m)(1,2,3,....,m)=(1,3,5,....,m  ,2,4,6,...m-1)
    which is again a cycle.
    Is this a suitable proof, and what else can I add to it.


    Thank You
    I consider this proof to be fine. You can add by showing the the LHS and the RHS agree for all values of 1,2,...,m in other words evaluating the LHS for any 1,2,...,m will give the same result as RHS and so the two functions (bijections) are the same.
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