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Math Help - Conjugacy Class of Elements of Order 4 in S8 and S12

  1. #1
    Super Member Bernhard's Avatar
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    Conjugacy Class of Elements of Order 4 in S8 and S12

    Dummit and Foote Section 4.3 Exercise 12 states:

    ================================================== ====
    Find a representative of each conjugacy class of elements of order 4 in S_8 and in S_{12}.
    ================================================== ====

    I am lost as to how to approach this problem.

    Can anyone please help?

    Peter
    Last edited by Bernhard; December 20th 2011 at 03:26 AM.
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  2. #2
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    Re: Conjugacy Class of Elements of Order 4 in S8 and S12

    well, first we have to have a way of describing the elements of order 4. let's tackle S8 first.

    clearly any 4-cycle is of order 4. since 4 = 2*2, the disjoint cycle decomposition of an element of S8 of order 4, can only contain 2-cycles and 4-cycles.

    that gives the following possibilities:

    (a b c d) a 4-cycle
    (a b c d)(e f) a disjoint 4-cycle and 2-cycle
    (a b c d)(e f)(g h) a 4-cycle and 2 disjoint 2-cycles
    (a b c d)(e f g h) 2 disjoint 4-cycles.

    in Sn, (disjoint) cycle types determine the conjugacy classes. that is two permutations with the same disjoint cycle types are conjugate, and vice-versa. can you tackle S12, now?
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Conjugacy Class of Elements of Order 4 in S8 and S12

    Thanks for that Deveno - most helpful

    Reckon I can tackle S12 OK now

    Peter
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