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Algebra, permutations.

  1. Feb 9, 2014 #1
    1. The problem statement, all variables and given/known data

    Suppose σ is an element of S_n, if σ^5=1, is σ necessarily odd or even

    2. Relevant equations

    [itex]Parity(\sigma)= (\sum_{i=1}^k(|c_i|-1)) mod 2[/itex]

    3. The attempt at a solution

    Really unsure how to even start this, I think I have to use the fact you can decompose every permutation into a product of disjoint or 2 cycles and use their properties to show in what cases it's even or odd but I'm not really sure.
     
  2. jcsd
  3. Feb 9, 2014 #2

    Dick

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    ##\sigma^5=\sigma \sigma \sigma \sigma \sigma=1##. ##1## is an even permutation. Suppose ##\sigma## were odd. Then what would be the parity of ##\sigma^5##?
     
  4. Feb 9, 2014 #3
    What does σ5 being even tell you?
     
    Last edited: Feb 9, 2014
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