Alternating Group on 4 Letters/Subgroups

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Subgroups 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 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


Order $2$ subgroups:

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


Order $3$ subgroups:

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


Order $4$ subgroup:

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


Examples of Subgroups

Order $3$

Let $P$ denote the subset of $A_4$:

$P := \set {e, a, p}$

Then $P$ is a subgroup of $A_4$.


Its left cosets are:

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


Its right cosets are:

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


Sources

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