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Math Help - Group Theory

  1. #1
    vogel
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    Group Theory

    Write down the group table for the premutuation group s3 and show that S3 is not abelian!!!!
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  2. #2
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    Quote Originally Posted by vogel View Post
    Write down the group table for the premutuation group s3 and show that S3 is not abelian!!!!
    Let,
    \sigma_1: 1\mapsto 1 , 2\mapsto 2, 3\mapsto 3.
    \sigma_2: 1\mapsto 2, 2\mapsto 3, 3\mapsto 1.
    \sigma_3: 1\mapsto 3, 2\mapsto 1, 3\mapsto 2.
    \sigma_4: 1\mapsto 2, 2\mapsto 1, 3\mapsto 3.
    \sigma_5: 1\mapsto 3, 2\mapsto 2, 3\mapsto 1.
    \sigma_6: 1\mapsto 1, 2\mapsto 3, 3\mapsto  2.

    Now if you let \alpha = \sigma_2 and \beta = \sigma_4. Then \{1,\alpha,\alpha^2 , \beta, \alpha \beta, \alpha^2 \beta \} expresses all S_3. Furthermore, confirm that \alpha \beta \not = \beta \alpha and you get what you wanted.
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