# Group Theory

• Feb 14th 2008, 07:44 AM
vogel
Group Theory
Write down the group table for the premutuation group s3 and show that S3 is not abelian!!!!(Doh)
• Feb 14th 2008, 08:34 AM
ThePerfectHacker
Quote:

Originally Posted by vogel
Write down the group table for the premutuation group s3 and show that S3 is not abelian!!!!(Doh)

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.