Klein Four-Group is Normal in A4

Theorem

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:

Consider the order $4$ subgroup $V$ of $A_4$, presented by Cayley table:

$\begin{array}{c|cccc} \circ & e & t & u & v \\ \hline e & e & t & u & v \\ t & t & e & v & u \\ u & u & v & e & t \\ v & v & u & t & e \\ \end{array}$

Then $V$ is normal in $A_4$.

Its index is:

$\index {A_4} V = \dfrac {\order {A_4} } {\order V} = \dfrac {12} 4 = 3$

The (left) cosets of $V$ are:

$V$

$A := a V$

$P := p V$

and the Cayley table of the quotient group $A_4 / V$ is given by:

$\begin{array}{c|ccc} \circ & V & A & P \\ \hline V & V & A & P \\ A & A & P & V \\ P & P & V & A \\ \end{array}$

Note that while $A_4 / V$ is Abelian, $A_4$ is not.

Proof

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$\index {A_4} V = 3$ follows from Lagrange's Theorem.

Proof 2

Conjugate subgroups have the same cardinality.

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Therefore, any conjugation of $V$ has $4$ elements.

By Subgroups of $A_4$, the only order-$4$ subgroup of $A_4$ is $V$.

Hence, any conjugation of $V$ in $A_4$ is $V$ itself.

By Subgroup equals Conjugate iff Normal, $V$ is normal in $A_4$.

$\blacksquare$

Sources

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.7$. Quotient groups: Example $127$