# 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:

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

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

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

### 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

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

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

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

## Subgroups

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

Trivial:

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

## Normality of Subgroups

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

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

## Conjugacy Classes

The conjugacy classes of $A_4$ are:

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