Alternating Group on 4 Letters/Subgroups/Examples/Order 3

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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}$


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:

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


Proof

We have that:

\(\ds a^2\) \(=\) \(\ds p\)
\(\ds a^3\) \(=\) \(\ds e\)

Thus $\set {e, a, p}$ forms a cyclic group generated by $a$.

Thus $\set {e, a, p}$ is a group which is a subset of $A_4$.

Hence by definition $\set {e, a, p}$ is a subgroup of $A_4$.


Then:

\(\ds t P\) \(=\) \(\ds \set {t \circ e, t \circ a, t \circ p}\)
\(\ds \) \(=\) \(\ds \set {t, b, q}\)


\(\ds u P\) \(=\) \(\ds \set {u \circ e, u \circ a, u \circ p}\)
\(\ds \) \(=\) \(\ds \set {u, c, r}\)


\(\ds v P\) \(=\) \(\ds \set {v \circ e, v \circ a, v \circ p}\)
\(\ds \) \(=\) \(\ds \set {v, d, s}\)


and:

\(\ds P t\) \(=\) \(\ds \set {e \circ t, a \circ t, p \circ t}\)
\(\ds \) \(=\) \(\ds \set {t, c, s}\)


\(\ds P u\) \(=\) \(\ds \set {e \circ u, a \circ u, p \circ u}\)
\(\ds \) \(=\) \(\ds \set {u, d, q}\)


\(\ds P v\) \(=\) \(\ds \set {e \circ v, a \circ v, p \circ v}\)
\(\ds \) \(=\) \(\ds \set {v, b, r}\)

$\blacksquare$


Sources

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