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:
\(\ds e\) | \(:=\) | \(\ds \text { the identity mapping}\) | ||||||||||||
\(\ds t\) | \(:=\) | \(\ds \tuple {1 2} \tuple {3 4}\) | ||||||||||||
\(\ds u\) | \(:=\) | \(\ds \tuple {1 3} \tuple {2 4}\) | ||||||||||||
\(\ds v\) | \(:=\) | \(\ds \tuple {1 4} \tuple {2 3}\) |
\(\ds a\) | \(:=\) | \(\ds \tuple {1 2 3}\) | ||||||||||||
\(\ds b\) | \(:=\) | \(\ds \tuple {1 3 4}\) | ||||||||||||
\(\ds c\) | \(:=\) | \(\ds \tuple {2 4 3}\) | ||||||||||||
\(\ds d\) | \(:=\) | \(\ds \tuple {1 4 2}\) |
\(\ds p\) | \(:=\) | \(\ds \tuple {1 3 2}\) | ||||||||||||
\(\ds q\) | \(:=\) | \(\ds \tuple {2 3 4}\) | ||||||||||||
\(\ds r\) | \(:=\) | \(\ds \tuple {1 2 4}\) | ||||||||||||
\(\ds s\) | \(:=\) | \(\ds \tuple {1 4 3}\) |
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:
\(\ds \) | \(\) | \(\ds \set e\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {a, b, c, d}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {p, q, r, s}\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {t, u, v}\) |