# Alternating Group on 4 Letters

## Group Example

Let $S_4$ denote the symmetric group on $4$ letters.

The alternating group on $4$ letters $A_4$ is the kernel of the mapping $\sgn: S_4 \to C_2$.

### Cycle Notation

It can be expressed in the form of permutations given in cycle notation as follows:

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

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

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

### Cayley Table

The Cayley table of $A_4$ can be written:

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

## Order of Elements

 $\ds \order e$ $=$ $\ds 1:$ Identity is Only Group Element of Order 1 $\ds \order t$ $=$ $\ds 2:$ $t^2 = e$ $\ds \order u$ $=$ $\ds 2:$ $u^2 = e$ $\ds \order v$ $=$ $\ds 2:$ $v^2 = e$

 $\ds \order a$ $=$ $\ds 3:$ $a^2 = p, a^3 = a p = e$ $\ds \order b$ $=$ $\ds 3:$ $b^2 = s, b^3 = b s = e$ $\ds \order c$ $=$ $\ds 3:$ $c^2 = q, c^3 = c q = e$ $\ds \order d$ $=$ $\ds 3:$ $d^2 = r, d^3 = d r = e$

 $\ds \order p$ $=$ $\ds 3:$ $p^2 = a, p^3 = p a = e$ $\ds \order q$ $=$ $\ds 3:$ $q^2 = c, q^3 = q c = e$ $\ds \order r$ $=$ $\ds 3:$ $r^2 = d, r^3 = r d = e$ $\ds \order s$ $=$ $\ds 3:$ $s^2 = b, s^3 = s b = e$

## Subgroups

The subsets of $A_4$ which form subgroups of $A_4$ are as follows:

Trivial:

 $\ds$  $\ds \set e$ Trivial Subgroup is Subgroup $\ds$  $\ds A_4$ Group is Subgroup of Itself
 $\ds$  $\ds \set {e, t}$ as $t^2 = e$ $\ds$  $\ds \set {e, u}$ as $u^2 = e$ $\ds$  $\ds \set {e, v}$ as $v^2 = e$
 $\ds$  $\ds \set {e, a, p}$ as $a^2 = p$, $a^3 = a p = e$ $\ds$  $\ds \set {e, b, s}$ as $b^2 = s$, $b^3 = b s = e$ $\ds$  $\ds \set {e, c, q}$ as $c^2 = q$, $c^3 = c q = e$ $\ds$  $\ds \set {e, d, r}$ as $d^2 = r$, $d^3 = d r = e$
 $\ds$  $\ds \set {e, t, u, v}$ Klein $4$-Group

## Normality of Subgroups

The normality status of the non-trivial proper subgroups of $A_4$ is as follows:

 $\ds T$ $:=$ $\ds \set {e, t}$ Not normal $\ds U$ $:=$ $\ds \set {e, u}$ Not normal $\ds V$ $:=$ $\ds \set {e, v}$ Not normal
 $\ds P$ $:=$ $\ds \set {e, a, p}$ Not normal $\ds Q$ $:=$ $\ds \set {e, c, q}$ Not normal $\ds R$ $:=$ $\ds \set {e, d, r}$ Not normal $\ds S$ $:=$ $\ds \set {e, b, s}$ Not normal
 $\ds K$ $:=$ $\ds \set {e, t, u, v}$ Normal

## Conjugacy Classes

The conjugacy classes of $A_4$ are:

 $\ds$  $\ds \set e$ $\ds$  $\ds \set {a, b, c, d}$ $\ds$  $\ds \set {p, q, r, s}$ $\ds$  $\ds \set {t, u, v}$