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Math Help - Help with presentation of the symmetric group in the simplest cases

  1. #1
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    iaoalst
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    Help with presentation of the symmetric group in the simplest cases

    The symmetric group on n-letters, Sn, may be described as follows.

    σi is the permutation that swaps the i:th element with the i+1 one.

    With the generators: σ1, . . ., σn-1

    And the following relations:

    • (σi)(σj+1)(σi) =(σj+1)(σi)(σj+1) ∀ i
    • σij = σjσi if |i - j| ≥ 2
    • (σi) = 1



    How do I verify the above for the two simplest cases where n = 2 and where n = 3?
    Can somebody help me out? I don't want anyone to solve this for me but rather if someone just could give me a hint and a push in the right direction.
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Help with presentation of the symmetric group in the simplest cases

    well the case n = 2 is fairly trivial, there is only one non-identity element, the permutation (1 2), of order 2. this is the only cyclic symmetric group, there is but one generator, σ1, and the only relation that applies is:

    1)2 = 1

    for the case n = 3, we have two generators: σ1 and σ2 (that is: (1 2) and (2 3)). the relation:

    σiσj = σjσ1 if |i - j| ≥ 2 also does not apply here (as |i - j| is either 0 or 1). i note in passing that what this captures is "disjoint cycles commute".

    so besides the fact that (σ1)2 = (σ2)2 = 1 (and this is always true for any transposition), the only non-trivial relation we have is:

    σ1σ2σ1 = σ2σ1σ2.

    note the LHS is the conjugate of σ2 by σ1, and the RHS is the conjugate of σ1 by σ2.

    that should be enough to get you started.
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