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Math Help - Theorem Question

  1. #1
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    Theorem Question

    The order of an element in S_n equals the LCM of the lengths of the cycles in its cycle decomposition.

    Suppose our cycle is (1 5 6)(2 3)(4). How can I use this to find the order of each element?
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  2. #2
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    Re: Theorem Question

    Quote Originally Posted by dwsmith View Post
    The order of an element in S_n equals the LCM of the lengths of the cycles in its cycle decomposition.

    Suppose our cycle is (1 5 6)(2 3)(4). How can I use this to find the order of each element?
    If a=(1 5 6)(2 3)(4) then ord(a) = LCM (ord(156), ord(23), ord(4)).

    Now, we have that ord(156)=3, ord(23)=2 and ord(4)=1. So, ord(a) =LCM(3,2,1) = 6.
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  3. #3
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    Re: Theorem Question

    you can actually see this in action, and it makes lots more sense:

    all 1-cycles are the identity, so we can disregard them (the map that takes 4-->4, and n-->n for all n ≠ 4, is the map that takes n-->n for all n).

    so (1 5 6)(2 3)(4) = (1 5 6)(2 3).

    [(1 5 6)(2 3)]^2 = (1 6 5) <--- here the 2-cycle factor "hits" the identity ( )
    [(1 5 6)(2 3)]^3 = (2 3) <--- here the 3-cycle factor is the identity
    [(1 5 6)(2 3)]^4 = (1 5 6) <--- 2-cycle is the identity
    [(1 5 6)(2 3)]^5 = (1 6 5)(2 3)
    [(1 5 6)(2 3)]^6 = ( ) <--- both factors are the identity.

    powers of disjoint cycles are easy to compute, because they (the disjoint cycles) commute with each other,

    (1 5 6)(2 3) = (2 3)(1 5 6).
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