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Question about permutations

  1. Oct 15, 2004 #1
    I have been working with the following question for quite awhile:

    Show that a permutation with an odd order must be an even
    permutation.

    I have made some progress, but I am having trouble putting it altogether
    to make my proof coherent.

    This is what i have so far:

    Let e= epsilon
    Say BA^(2ka+1)= ae. Then BA^(2ka)=BA^(-1).
    But BA^(2k)=(BA^ka)^2 is even.

    I know that I am on the right track but I can't seem to put
    it altogether. Can someone help me please. If I could
    just have it explained Iam sure I will understand.
     
  2. jcsd
  3. Oct 16, 2004 #2

    HallsofIvy

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    It would help if you told us what your notation, etc. meant.

    1. How are you defining the "order" of a permutation?

    2. Are A and B premutations? If so which is intended to be the "permutation with odd order?

    3. What is k? what is a?
     
  4. Oct 17, 2004 #3

    mathwonk

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    I suppose the order of a permutation is its order as an element of the group of permutations, i.e. that A^k = id for some odd number k>0, and for no smaller positive integer.

    Then we claim A is "even". Recall that a product of an odd number of "odd" permutations is also "odd"...........

    does that help?
     
  5. Oct 18, 2004 #4
    Yes you have helped me very much, I think I have a handle on the problem know than you both. :)
     
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