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Permutation Groups

  1. Jul 25, 2010 #1
    1. The problem statement, all variables and given/known data
    This is a problem from a chapter entitled "Permutation Groups" of an abstract algebra text.
    1. Let α = ( 1 3 5 7 ) and β = (2 4 8) o (1 3 6) ∈ S8 Find α o β o α-1.
    2. Let α = ( 1 3) o (5 8) and β = (2 3 6 7) ∈ S8 Find α o β o α-1.


    2. Relevant equations
    Sn is the set of all permutations on In, where In={1,2,3,...,n}

    Also, o is known to be associative, but not commutative.
    α and β are conjugate if there exists γ ∈ Sn such that γ o α o γ-1 = β

    Then, let π = (i1 i2 ... il) ∈ Sn be a cycle. Then for all α ∈ Sn, α o π o α-1 = (α(i1) α(i2) ... α(il))

    3. The attempt at a solution
    I was able to calculate other problems easy enough that did not contain the composition of permutation cycles. Also, I can write the composition as a two row notation instead of a cycle, but then I don't know which elements I use when I calculate the conjugate against alpha. If I left the beta as a composition, maybe I could use the associative property and apply one element first, but I'm at a loss.

    A worked out solution would be really great- my professor assigned me 11 of these calculations, saying that they would be "really easy" so this many will not be a big deal. Thanks, prof.
     
  2. jcsd
  3. Jul 25, 2010 #2

    lanedance

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    Homework Helper

    so for the first, note that a will commute with the first cycle of beta

    you will also have to have a think about what the inverse of a given cylce is

    a nice way to do these problems is as folllows, say you want to examine (23)o(12)which don't commute, but write it out as follows:
    start = 1-2-3-4-5-6-7-8
    (12) = 2-1-3-4-5-6-7-8
    (23) = 2-3-1-4-5-6-7-8 = (123)

    note be careful exactly how the cycle is interpreted as they can be written in different ways, I read (123) as 2 goes to 1, 3 goes to 2, and then 1 goes to 3
     
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