Question

Prove that $\mathit{N}\mathbf{=}\mathbf{\{}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{,}\mathbf{\left(}\mathbf{12}\mathbf{\right)}\mathbf{\left(}\mathbf{34}\mathbf{\right)}\mathbf{,}\mathbf{\left(}\mathbf{13}\mathbf{\right)}\mathbf{\left(}\mathbf{24}\mathbf{\right)}\mathbf{,}\mathbf{\left(}\mathbf{14}\mathbf{\right)}\mathbf{\left(}\mathbf{23}\mathbf{\right)}\mathbf{\}}$ is a normal subgroup of${\mathit{A}}_{\mathbf{4}}$ . Hence,${\mathit{A}}_{\mathbf{4}}$ is not simple.

Short answer

It is proved that,$N=\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\}$ is a normal subgroup of $A_4$.

Step-by-step solution

Determine N={(1),(12)(34),(13)(24),(14)(23)}

Consider the given subgroup,$N=\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\}$.

$N=\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\}$is a subgroup of $A_4$.

The elements of $A_4$other than the elements in $N$are .

$\{\left(123\right),\left(132\right),\left(124\right),\left(142\right),\left(134\right),\left(143\right),\left(234\right),\left(243\right)\}$

Determine table

$ng$	$\left(123\right)$	role="math" localid="1654606881251" $\left(132\right)$	role="math" localid="1654606904505" $\left(124\right)$	$\left(142\right)$	$\left(134\right)$	$\left(143\right)$	$\left(234\right)$	$\left(243\right)$
$\left(1\right)$	$\left(1\right)$	$\left(1\right)$	$\left(1\right)$	$\left(1\right)$	$\left(1\right)$	$\left(1\right)$	$\left(1\right)$	$\left(1\right)$
$\left(12\right)\left(34\right)$	$\left(14\right)\left(23\right)$	$\left(13\right)\left(24\right)$	$\left(13\right)\left(24\right)$	$\left(14\right)\left(23\right)$	$\left(14\right)\left(23\right)$	$\left(13\right)\left(24\right)$	$\left(13\right)\left(24\right)$	$\left(14\right)\left(23\right)$
$\left(13\right)\left(24\right)$	$\left(12\right)\left(34\right)$	$\left(14\right)\left(23\right)$	$\left(14\right)\left(23\right)$	$\left(12\right)\left(34\right)$	$\left(12\right)\left(34\right)$	$\left(14\right)\left(23\right)$	$\left(14\right)\left(23\right)$	$\left(12\right)\left(34\right)$
$\left(14\right)\left(23\right)$	$\left(13\right)\left(24\right)$	$\left(12\right)\left(34\right)$	$\left(12\right)\left(34\right)$	$\left(13\right)\left(24\right)$	$\left(13\right)\left(24\right)$	$\left(12\right)\left(34\right)$	$\left(12\right)\left(34\right)$	$\left(13\right)\left(24\right)$

The above table shows the result of $gn{g}^{-1}$and here,$g$ is one of these 3-cycles and $n\in N$.

Then, it is clear that in all cases $gn{g}^{-1}\in N$, so that $N$is normal.

Hence,$N=\{\left(1\right),\left(12\right)\left(34\right),\left(13\right)\left(24\right),\left(14\right)\left(23\right)\}$ is a normal subgroup of $A_4$.