Alternating Group on 4 Letters

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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


Order $2$ subgroups:

\(\displaystyle \) \(\) \(\displaystyle \set {e, t}\) as $t^2 = e$
\(\displaystyle \) \(\) \(\displaystyle \set {e, u}\) as $u^2 = e$
\(\displaystyle \) \(\) \(\displaystyle \set {e, v}\) as $v^2 = e$


Order $3$ subgroups:

\(\displaystyle \) \(\) \(\displaystyle \set {e, a, p}\) 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 $d^2 = r$, $d^3 = d r = e$


Order $4$ subgroup:

\(\displaystyle \) \(\) \(\displaystyle \set {e, t, u, v}\) Klein $4$-Group


Normality of Subgroups

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


Order $2$ subgroups:

\(\displaystyle T\) \(:=\) \(\displaystyle \set {e, t}\) Not normal
\(\displaystyle U\) \(:=\) \(\displaystyle \set {e, u}\) Not normal
\(\displaystyle V\) \(:=\) \(\displaystyle \set {e, v}\) Not normal


Order $3$ subgroups:

\(\displaystyle P\) \(:=\) \(\displaystyle \set {e, a, p}\) 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


Order $4$ subgroup:

\(\displaystyle K\) \(:=\) \(\displaystyle \set {e, t, u, v}\) Normal


Conjugacy Classes

The conjugacy classes of $A_4$ are:

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