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Let σ a cycle of length s. Then σ^2 is a cycle iff s is odd.

  1. Jun 20, 2012 #1
    Proposition: Suppose σ is a cycle of length s. Then σ2 is a cycle if and only if s is odd.

    Quick remark... The following "proof" seems more like an explanation. This is for a self-study and I am looking back through my notes to refine/correct/improve. If you were correcting this, say out of 10pts, what would you give it?

    Proof:

    Let σ be a cycle of length s. That is, σ=(a1a2a3....as-1as). Suppose s is odd. Now, we calculate σ2 and we obtain that σ2=(a1a3a5a7....asa2a4a6....as-1). We notice that all odd numbered terms appear first in our cycle. Then, as, our last odd numbered term, gets sent to a2. The term a2 starts a sequence of moving even numbered terms to the next even numbered term, which concludes with as-1, which is also even. The term as-1 gets sent to a1 which completes the cycle. Now if s was even, then the term as-1 would be odd. But, as-1 would still get sent to a1. Hence, σ2 would be written as the product of disjoint cycles. That is,
    [Equation 1] σ2=(a1a3a5a7....as-1)(a2a4a6....as). Thus, σ2 would not be a cycle.

    Similarly, if we have that σ2 is cycle, then we deduce that s must be odd or else a situation similar to [Equation 1] would arise.

    QED


    Thanks for the help!! :)
     
  2. jcsd
  3. Jun 21, 2012 #2

    tiny-tim

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    hi jmjlt88! :smile:

    sorry, but personally i'd give you 6, since it's a bit confusing and, although i think you've probably grasped the principles involved, i'm not convinced you have :redface:

    since you apparently see how to make the cycles, it would be simpler just to show the single cycle and the pair of cycles, for the two cases

    (also, i recommend using 2n and 2n+1, instead of the same symbol for both odd and even numbers :wink:)
     
  4. Jun 21, 2012 #3
    Hey tiny-tim! Thanks! I actually did it like you said on paper. In my notes, I wrote the cycle in different notation [the two line notation where you put the elements on top and what they get sent to in the botton] which made it clearer. I then just observed what happened when s was odd, and then when s was even. Maybe if you saw that, I'd get a passing mark!! =)
     
  5. Jun 22, 2012 #4

    tiny-tim

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    hey jmjlt88! :smile:

    (just got up :zzz:)

    yes that looks better :smile:

    except i'm worried about …
    … if by "observed" you mean explained in words, it would be clearer (and probably more convincing) if you just did it with maths, eg (2k 2(k+1) etc) and (2k+1 2(k+1)+1 etc) :wink:
     
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