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Math Help - conjugacy class and class equation

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    conjugacy class and class equation

    Describe the conjugacy class and write the class equation for {Z}_{3} x {S}_{3}.
    {S}_{3} here being the symmetries of the equilateral triangle.
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    Quote Originally Posted by frankdent1 View Post
    Describe the conjugacy class and write the class equation for {Z}_{3} x {S}_{3}.
    {S}_{3} here being the symmetries of the equilateral triangle.
    Let a=(123),b=(12) then S_3 = \{ e,a,a^2,b,ab,a^2b\}.
    The conjugacy classes for S_3 are: \{e\}, \{a,a^2\}, \{b,ab,a^2b\}.
    The conjugacy classes for \mathbb{Z}_3 are: \{0\},\{1\},\{2\}.

    Therefore the conjugacy classes for \mathbb{Z}_3\times S_3 are I\times J where I is a conjugacy class of \mathbb{Z}_3 and J is a conjugacy class of \mathbb{S}_3. Note, there are total of 3\cdot 3=9 conjugacy classes for \mathbb{Z}_3\times S_3. Furthermore, |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: 18 = 1 + 1 + 1 + 2 + 2 + 2 + 3 + 3 + 3.
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