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Math Help - conjugacy classes on symmetric groups

  1. #1
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    conjugacy classes on symmetric groups

    dear math help forums,

    i fear that with the advent of my upcoming abstract algebra exam, the world shall end. however, in order to fend off the end of the world, i shall try my best to defeat this exam, and save the planet. your help in my triumph over this exam will be greatly appreciated!

    i'm struggling with the following problem:

    Find all the conjugacy classes of elements in the symmetric group S4.

    i know that since S4 is not abelian, the conjugacy classes will not be singletons. wikipedia tells me the following:

    The symmetric group S4, consisting of all 24 permutations of four elements, has five conjugacy classes, listed with their orders:

    • no change (1)
    • interchanging two (6)
    • a cyclic permutation of three (8)
    • a cyclic permutation of all four (6)
    • interchanging two, and also the other two (3)

    however, the numbers after these statements are mystifying to me. any help in explaining how these numbers came into fruition would be crucial to saving our home Earth.
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  2. #2
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    Quote Originally Posted by chrisneedshelp View Post
    Find all the conjugacy classes of elements in the symmetric group S4.
    Remember if \sigma \in S_n then we can write \sigma = \theta_1 \theta_2 ... \theta_r and r\geq 1 where \theta_j are cycles so that \theta_i and \theta_j are disjoint for i\not = j. Two elements \sigma,\tau \in S_n are said to have the same structure iff when we write \sigma = \theta_1 \theta_2 ... \theta_r and \tau = \eta_1 \eta_2 ... \eta_s then r=s and can be relabled so that \theta_j has the same cycle length as \eta_j. Two elements \sigma,\tau are conjugate to eachother if and only if \sigma and \tau have the same structure.

    If you take \sigma \in S_4 then we have a certain number of possibilities:
    • \sigma is 1-cycle i.e. identity --> there is only one element that has this property
    • \sigma is a 2-cycle --> there are {4\choose 2} = 6 such elements
    • \sigma is a 3-cycle --> there are 2! {4\choose 3} = 8 such elements
    • \sigma is a 4-cycle --> there are 3! = 6 such elements
    • \sigma is a product of two 2-cycles --> there are \frac{1}{2}{4\choose 2} = 3 such elements


    As a check confirm that 1+3+6+6+8 = 24 (this happens to be the conjugacy class equation).
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    • \sigma is 1-cycle i.e. identity --> there is only one element that has this property
    • \sigma is a 2-cycle --> there are {4\choose 2} = 6 such elements
    • \sigma is a 3-cycle --> there are 2! {4\choose 3} = 8 such elements
    • \sigma is a 4-cycle --> there are 3! = 6 such elements
    • \sigma is a product of two 2-cycles --> there are \frac{1}{2}{4\choose 2} = 3 such elements
    The 1 cycle identity is just (1,2,3,4) followed by (1,2,3,4).
    the 2 cycles would be all permutations of moving elements once? Is this correct? I'm not sure about these values, myself. I've probably missed something along the way that makes these values make sense.

    thanks for your reply
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  4. #4
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    Quote Originally Posted by chrisneedshelp View Post
    The 1 cycle identity is just (1,2,3,4) followed by (1,2,3,4).
    the 2 cycles would be all permutations of moving elements once? Is this correct? I'm not sure about these values, myself. I've probably missed something along the way that makes these values make sense.

    thanks for your reply
    Here are the conjugacy classes:
    \{ \text{id} \}
    \{ (12)(34), (13)(24),(14)(23) \}
    \{ (12),(13),(14),(23),(24),(34) \}
    \{(123),(124),(132),(134),(142),(143),(234),(243)\  }
    \{ (1234),(1243),(1324),(1343),(1423),(1432)\}
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  5. #5
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    Got it! thanks a lot for your help. the earth will survive another day
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