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Math Help - Permutation group order

  1. #1
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    Permutation group order

    Show that A_{8} contains an element of order 15.

    Note: A_{8} is the group of even permutation of 8.

    I really don't know how to start this proof, please help.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Show that A_{8} contains an element of order 15.

    Note: A_{8} is the group of even permutation of 8.

    I really don't know how to start this proof, please help.
    Let \sigma be a cycle with length n. And let \tau be a cycle with length m. Then the order of the permutation \sigma \tau is equal to \mbox{lcm}(n,m). So the idea is to find an even cycle of length 5 and another even cycle with length 3 and take their product.
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  3. #3
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    Quote Originally Posted by Bernhard Riemann
    Let \sigma be a cycle with length n. And let \tau be a cycle with length m. Then the order of the permutation \sigma \tau is equal to \mbox{lcm}(n,m). So the idea is to find an even cycle of length 5 and another even cycle with length 3 and take their product.
    Okay how about I give you my example and you confirm the details.

    1)Consider (1,2,3) = (1,2)(2,3) this is an even permutation, why? So (1,2,3)\in A_8.

    2)Consider (1,2,3,4,5) = (1,2)(2,3)(3,4)(4,5) this is an even permutation, why? So (1,2,3,4,5)\in A_8.

    3)Define \phi = (1,2,3)(1,2,3,4,5) this is an even permutation, why? So \phi \in A_8.

    4)By my previous comment \mbox{ord}(\phi) = \mbox{lcm}(3,5) = 15.
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  4. #4
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    So I pick O = (1 2 3) and r = (4 5 6 7 8 9), they are both even permutations, and O has length 3, r has length 5, so ord(Or) = 15.

    Thank you!
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