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Math Help - Permutations (Want to check my answer)

  1. #1
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    Permutations (Want to check my answer)

    I just want to make sure my answer is correct


    Questions: List all even permutations of symmetry group S4


    Answer: There are 12 even permutations in S4
    (1234)(1342)(1423)(2143)(3124)(4132)(2431)(2314)(4 213)(3241)(3412)(4321)


    Please Help
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  2. #2
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    Quote Originally Posted by mathlg
    I just want to make sure my answer is correct


    Questions: List all even permutations of symmetry group S4


    Answer: There are 12 even permutations in S4
    (1234)(1342)(1423)(2143)(3124)(4132)(2431)(2314)(4 213)(3241)(3412)(4321)


    Please Help
    Rememeber that the cardinality of the finite 'alternating group" is \frac{n!}{2}.
    In this case n=4 thus,
    \frac{4!}{2}=\frac{24}{2}=12
    Thus there are 12 even premutations.

    For the second part, all the premutations you named are odd!
    Because,
    (a,b,c,d)=(a,b)(b,c)(c,d)
    The number of transpositions here is an odd number!
    Also, the premutation
    (a,b,c,d)=(d,a,b,c)

    Thus, there are two mistakes, first they are all odd

    and not all are distinct-which basically means you did not list all the odd premutations.
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  3. #3
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    A4 = {(1,2,3), (1,3,2), (1,2,4), (1,4,2), (1,3,4), (1,4,3), (2,3,4), (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3),e}.
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