# conjugacy class and class equation

• Mar 1st 2009, 11:23 AM
frankdent1
conjugacy class and class equation
Describe the conjugacy class and write the class equation for $\displaystyle {Z}_{3}$ x $\displaystyle {S}_{3}$.
$\displaystyle {S}_{3}$ here being the symmetries of the equilateral triangle.
• Mar 1st 2009, 06:52 PM
ThePerfectHacker
Quote:

Originally Posted by frankdent1
Describe the conjugacy class and write the class equation for $\displaystyle {Z}_{3}$ x $\displaystyle {S}_{3}$.
$\displaystyle {S}_{3}$ here being the symmetries of the equilateral triangle.

Let $\displaystyle a=(123),b=(12)$ then $\displaystyle S_3 = \{ e,a,a^2,b,ab,a^2b\}$.
The conjugacy classes for $\displaystyle S_3$ are: $\displaystyle \{e\}, \{a,a^2\}, \{b,ab,a^2b\}$.
The conjugacy classes for $\displaystyle \mathbb{Z}_3$ are: $\displaystyle \{0\},\{1\},\{2\}$.

Therefore the conjugacy classes for $\displaystyle \mathbb{Z}_3\times S_3$ are $\displaystyle I\times J$ where $\displaystyle I$ is a conjugacy class of $\displaystyle \mathbb{Z}_3$ and $\displaystyle J$ is a conjugacy class of $\displaystyle \mathbb{S}_3$. Note, there are total of $\displaystyle 3\cdot 3=9$ conjugacy classes for $\displaystyle \mathbb{Z}_3\times S_3$. Furthermore, $\displaystyle |I\times J| = |J|$, so there are three conjugacy classes that has 1 element, three conjugacy classes that have 2 elements, three conjugacy classes that have 3 elements.

Thus, the class equation is: $\displaystyle 18 = 1 + 1 + 1 + 2 + 2 + 2 + 3 + 3 + 3$.