Alternating Group on 4 Letters/Normality of Subgroups

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


Order $2$ subgroups:

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


Order $3$ subgroups:

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


Order $4$ subgroup:

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


Proof

Testing one of the left cosets of $T = \set {e, t}$ against its corresponding right coset:

\(\ds a T\) \(=\) \(\ds \set {a \circ e, a \circ t}\)
\(\ds \) \(=\) \(\ds \set {a, c}\)
\(\ds T a\) \(=\) \(\ds \set {e \circ a, t \circ a}\)
\(\ds \) \(=\) \(\ds \set {a, b}\)
\(\ds \) \(\ne\) \(\ds a T\)

The left coset does not equal the right coset and so $T$ is not normal in $A_4$.

$\Box$


Testing one of the left cosets of $U = \set {e, u}$ against its corresponding right coset:

\(\ds b U\) \(=\) \(\ds \set {b \circ e, b \circ u}\)
\(\ds \) \(=\) \(\ds \set {b, c}\)
\(\ds U b\) \(=\) \(\ds \set {e \circ b, u \circ b}\)
\(\ds \) \(=\) \(\ds \set {b, d}\)
\(\ds \) \(\ne\) \(\ds b U\)

The left coset does not equal the right coset and so $U$ is not normal in $A_4$.

$\Box$


Testing one of the left cosets of $V = \set {e, v}$ against its corresponding right coset:

\(\ds c V\) \(=\) \(\ds \set {c \circ e, c \circ v}\)
\(\ds \) \(=\) \(\ds \set {c, d}\)
\(\ds V c\) \(=\) \(\ds \set {e \circ c, v \circ c}\)
\(\ds \) \(=\) \(\ds \set {c, b}\)
\(\ds \) \(\ne\) \(\ds c V\)

The left coset does not equal the right coset and so $V$ is not normal in $A_4$.

$\Box$


Testing one of the left cosets of $P = \set {e, a, p}$ against its corresponding right coset:

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

The left coset does not equal the right coset and so $P$ is not normal in $A_4$.

$\Box$


Testing one of the left cosets of $Q = \set {e, c, q}$ against its corresponding right coset:

\(\ds t Q\) \(=\) \(\ds \set {t \circ e, t \circ c, t \circ q}\)
\(\ds \) \(=\) \(\ds \set {t, d, p}\)
\(\ds Q t\) \(=\) \(\ds \set {e \circ t, c \circ t, q \circ t}\)
\(\ds \) \(=\) \(\ds \set {t, a, r}\)
\(\ds \) \(\ne\) \(\ds t Q\)

The left coset does not equal the right coset and so $Q$ is not normal in $A_4$.

$\Box$


Testing one of the left cosets of $R = \set {e, d, r}$ against its corresponding right coset:

\(\ds t R\) \(=\) \(\ds \set {t \circ e, t \circ d, t \circ r}\)
\(\ds \) \(=\) \(\ds \set {t, c, s}\)
\(\ds R t\) \(=\) \(\ds \set {e \circ t, d \circ t, r \circ t}\)
\(\ds \) \(=\) \(\ds \set {t, b, q}\)
\(\ds \) \(\ne\) \(\ds t R\)

The left coset does not equal the right coset and so $R$ is not normal in $A_4$.

$\Box$


Testing one of the left cosets of $S = \set {e, b, s}$ against its corresponding right coset:

\(\ds t S\) \(=\) \(\ds \set {t \circ e, t \circ b, t \circ s}\)
\(\ds \) \(=\) \(\ds \set {t, a, r}\)
\(\ds S t\) \(=\) \(\ds \set {e \circ t, b \circ t, s \circ t}\)
\(\ds \) \(=\) \(\ds \set {t, d, p}\)
\(\ds \) \(\ne\) \(\ds t S\)

The left coset does not equal the right coset and so $S$ is not normal in $A_4$.

$\Box$


The cosets of $K = \set {e, t, u, v}$ are as follows:

\(\ds a K\) \(=\) \(\ds \set {a \circ e, a \circ t, a \circ u, a \circ v}\)
\(\ds \) \(=\) \(\ds \set {a, c, d, b}\)
\(\ds K a\) \(=\) \(\ds \set {e \circ a, t \circ a, u \circ a, v \circ a}\)
\(\ds \) \(=\) \(\ds \set {a, b, c, d}\)
\(\ds \) \(=\) \(\ds a K\)


\(\ds p K\) \(=\) \(\ds \set {p \circ e, p \circ t, p \circ u, p \circ v}\)
\(\ds \) \(=\) \(\ds \set {p, s, q, r}\)
\(\ds K p\) \(=\) \(\ds \set {e \circ p, t \circ p, u \circ p, v \circ p}\)
\(\ds \) \(=\) \(\ds \set {p, q, r, s}\)
\(\ds \) \(=\) \(\ds p K\)

The left cosets equal the right cosets and so $K$ is normal in $A_4$.

$\Box$


Sources

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