Alternating Group on 4 Letters/Conjugacy Classes

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Conjugacy Classes of the Alternating Group on 4 Letters

Let $A_4$ denote the Alternating Group on 4 Letters, whose Cayley table is given as:

$\begin{array}{c|cccc|cccc|cccc} \circ & e & t & u & v & a & b & c & d & p & q & r & s \\ \hline e & e & t & u & v & a & b & c & d & p & q & r & s \\ t & t & e & v & u & b & a & d & c & q & p & s & r \\ u & u & v & e & t & c & d & a & b & r & s & p & q \\ v & v & u & t & e & d & c & b & a & s & r & q & p \\ \hline a & a & c & d & b & p & r & s & q & e & u & v & t \\ b & b & d & c & a & q & s & r & p & t & v & u & e \\ c & c & a & b & d & r & p & q & s & u & e & t & v \\ d & d & b & a & c & s & q & p & r & v & t & e & u \\ \hline p & p & s & q & r & e & v & t & u & a & d & b & c \\ q & q & r & p & s & t & u & e & v & b & c & a & d \\ r & r & q & s & p & u & t & v & e & c & b & d & a \\ s & s & p & r & q & v & e & u & t & d & a & c & b \\ \end{array}$


The conjugacy classes of $A_4$ are:

\(\displaystyle \) \(\) \(\displaystyle \set e\)
\(\displaystyle \) \(\) \(\displaystyle \set {a, b, c, d}\)
\(\displaystyle \) \(\) \(\displaystyle \set {p, q, r, s}\)
\(\displaystyle \) \(\) \(\displaystyle \set {t, u, v}\)


Proof

The elements of the Alternating Group on 4 Letters can be expressed in cycle notation thus:

\(\displaystyle e\) \(:=\) \(\displaystyle \text { the identity mapping}\)
\(\displaystyle t\) \(:=\) \(\displaystyle \tuple {1 2} \tuple {3 4}\)
\(\displaystyle u\) \(:=\) \(\displaystyle \tuple {1 3} \tuple {2 4}\)
\(\displaystyle v\) \(:=\) \(\displaystyle \tuple {1 4} \tuple {2 3}\)


\(\displaystyle a\) \(:=\) \(\displaystyle \tuple {1 2 3}\)
\(\displaystyle b\) \(:=\) \(\displaystyle \tuple {1 3 4}\)
\(\displaystyle c\) \(:=\) \(\displaystyle \tuple {2 4 3}\)
\(\displaystyle d\) \(:=\) \(\displaystyle \tuple {1 4 2}\)


\(\displaystyle p\) \(:=\) \(\displaystyle \tuple {1 3 2}\)
\(\displaystyle q\) \(:=\) \(\displaystyle \tuple {2 3 4}\)
\(\displaystyle r\) \(:=\) \(\displaystyle \tuple {1 2 4}\)
\(\displaystyle s\) \(:=\) \(\displaystyle \tuple {1 4 3}\)



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