# Group Theory

• Feb 14th 2008, 06:44 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, 07:34 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,
$\displaystyle \sigma_1: 1\mapsto 1 , 2\mapsto 2, 3\mapsto 3$.
$\displaystyle \sigma_2: 1\mapsto 2, 2\mapsto 3, 3\mapsto 1$.
$\displaystyle \sigma_3: 1\mapsto 3, 2\mapsto 1, 3\mapsto 2$.
$\displaystyle \sigma_4: 1\mapsto 2, 2\mapsto 1, 3\mapsto 3$.
$\displaystyle \sigma_5: 1\mapsto 3, 2\mapsto 2, 3\mapsto 1$.
$\displaystyle \sigma_6: 1\mapsto 1, 2\mapsto 3, 3\mapsto 2$.

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