# Math Help - Group Theory

1. ## Group Theory

Write down the group table for the premutuation group s3 and show that S3 is not abelian!!!!

2. Originally Posted by vogel
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.