Abstract Algebra: Theory and Applications · Problem 14.5.6

The Sylow Theorems — Thomas Judson

Mathematics — Abstract Algebra · conjugacy-classes-class-equation

Problem

Find the conjugacy classes and the class equation for each of the following groups. (a) S4S_4 (b) D5D_5 (c) Z9\mathbb{Z}_9 (d) Q8Q_8 Hint. The conjugacy classes for S4S_4 are O(1)={(1)},O(12)={(1 2),(1 3),(1 4),(2 3),(2 4),(3 4)},O(1 2)(3 4)={(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)},O(123)={(1 2 3),(1 3 2),(1 2 4),(1 4 2),(1 3 4),(1 4 3),(2 3 4),(2 4 3)},O(1234)={(1 2 3 4),(1 2 4 3),(1 3 2 4),(1 3 4 2),(1 4 2 3),(1 4 3 2)}.\begin{aligned} \mathcal{O}_{(1)} &= \{(1)\},\\ \mathcal{O}_{(12)} &= \{(1\ 2),(1\ 3),(1\ 4),(2\ 3),(2\ 4),(3\ 4)\},\\ \mathcal{O}_{(1\ 2)(3\ 4)} &= \{(1\ 2)(3\ 4),(1\ 3)(2\ 4),(1\ 4)(2\ 3)\},\\ \mathcal{O}_{(123)} &= \{(1\ 2\ 3),(1\ 3\ 2),(1\ 2\ 4),(1\ 4\ 2),(1\ 3\ 4),(1\ 4\ 3),(2\ 3\ 4),(2\ 4\ 3)\},\\ \mathcal{O}_{(1234)} &= \{(1\ 2\ 3\ 4),(1\ 2\ 4\ 3),(1\ 3\ 2\ 4),(1\ 3\ 4\ 2),(1\ 4\ 2\ 3),(1\ 4\ 3\ 2)\}. \end{aligned} The class equation is 1+3+6+6+8=241 + 3 + 6 + 6 + 8 = 24.

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