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Thread: conjugacy class and class equation

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by frankdent1 View Post
    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: $\displaystyle 18 = 1 + 1 + 1 + 2 + 2 + 2 + 3 + 3 + 3$.
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