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Math Help - Left and Right Cosets

  1. #1
    Junior Member
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    Left and Right Cosets

    Let G = S3, the symmetric group of degree 3 and let H = {i,f} where f(x1) = x2, f(x2) = x1, f(x3) = x3
    a) find all the left cosets of H in G
    b) find all the right cosets of H in G
    c) Is every left coset of H a right coset of H?

    Please show explicit steps I'm very confused on how to prove these coset problems. Thanks so much!
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  2. #2
    Super Member Gamma's Avatar
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    cycle notation

    I find it a lot easier to use cycle notation when working with the symmetric group. example (1,2,4, 7)\in S_7 is the permutation that takes 1 to 2, 2 to 4, 4 to 7 and 7 to 1. The unmentioned numbers are left the same.
    Your coset H=\{(1), (1,2)\} it has size two and |S_n|=n! \Rightarrow |S_3|=3!=6. So you should expect 3 cosets each of size 2. Basically you just gotta multiply them out and see what happens.
    Here are the left cosets of H
    (1)H=\{(1), (1,2)\}
    (1,3)H=\{(1,3), (1,2,3)\}
    (2,3)H=\{(2,3), (1,3,2)\}

    Right cosets of H found similarly
    H(1)=\{(1), (1,2)\}
    H(1,3)=\{(1,3), (1,3,2)\}
    H(2,3)=\{(2,3), (1,2,3)\}

    Compare these cosets and see that the last two do not match up, so these are not the same. In particular this tells you that H is infact not a normal subgroup of S_3 because (1,3)H \not = H(1,3)
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