Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Clarification of Permuatation Question

  1. Oct 16, 2004 #1
    NEED HELP!!! with permutation proof

    Sorry for any confusion the question I have is:

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

    The order of a permutation of a finite set written in disjoint cycle form
    is the least common multiple of the lengths of the cycles.

    This is what I have worked out so far:

    Let e = epsilon
    Let B = a permutation
    Let k = any integer

    Now say B^(2k+1) = e. Where B^(2k+1) is an odd permutation.
    Then B^(2k)= B^(-1).
    But B^(2k) = B^(k)^2 is even.

    I would really appreciate some help in putting this
    proof together in a more coherent fashion.
    I am very confused. Thanks
    Last edited: Oct 17, 2004
  2. jcsd
  3. Oct 18, 2004 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    More important than that surely is the definition of the sign (even or odd) at no point do you use its properties so surely something must be telling you you need some more details.

    sign is multiplicative: sign(xy)=sign(x)sign(y)

    so it suffices to show that an element of odd order cannot possess any cycles of odd sign (when written as of disjoint cycles), which is where you're order being the lcm comes in
  4. Oct 19, 2004 #3
    Thanks to Matt for all your help. I understand the problem better know and have been able to solve it. I'm very grateful
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook