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Intro to Algebra, Symmetric group # of elements of order 4 in S6

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data

    How many elements of order 4 are in S6? (symmetric group with order 6)

    2. Relevant equations



    3. The attempt at a solution

    So, the different forms of elements with order 4 in S6 are

    (abcd)(ef), (abcd)

    from there im sunk on how to calculate. I know there are 6! = 720 total elements in S6.
     
  2. jcsd
  3. Dec 14, 2011 #2

    Deveno

    User Avatar
    Science Advisor

    well, first, how many 4-cycles are there?

    first, we have to pick 4 elements to permute out of 6. how many ways are there to choose 4 elements out of 6?

    let's say we've picked our 4: {a,b,c,d}.

    how many distinct 4-cycles can you make on that set (this should be the same number of 4-cycles as we have in S4, right?).

    that should help you with the "pure" 4 -cycles.

    now, suppose you have (a b c d)(e f).

    once we've chosen a,b,c,d (step one in counting the 4-cycles), do we really have any choice as how to choose e and f? and given any two elements of {1,2,3,4,5,6}, say, j and k, how many transpositions can be made from j and k? if you think about this the right way, you don't even have to count the 4+2 cycles.
     
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