# Alternating Group on 4 Letters/Conjugacy Classes

## 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:

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

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

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