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Abstract Algebra: Parity of a Permutation

  1. Jan 19, 2012 #1
    1. The problem statement, all variables and given/known data

    How do I determine the parity of a permutation? I think my reasoning may be faulty.

    By a theorem, an n-cycle is the product of (n-1) transpositions. For example, a 5 cycle can be written as 4 transpositions.

    Now say I have a permutation written in cycle notation: (1 4 5)(2 3).

    I say it is odd, because (1 4 5) can be written as two transpositions, and (2 3) is already a transposition, giving 3 total transpositions:

    (1 4 5)(2 3) = (1 5)(1 4)(2 3).

    Since the number of transpositions is odd, the permutation must be odd.
    Agree or disagree?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 19, 2012 #2

    jbunniii

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    Agree. Note that you are implicitly using a very important and non-trivial theorem, which is that even though a permutation may be written in many different ways as a product of transpositions, the parity is the same no matter which product you choose.

    For example, a 5-cycle may be written as a product of 4 transpositions, or 6 transpositions, or 8, etc., but there's no way to write it as a product of 5 or 7 or 9...
     
  4. Jan 19, 2012 #3
    Thank you, for the swift reply.
     
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