Question

Prove that $\mathit{N}\mathbf{=}\mathbf{\{}{\mathit{r}}_{\mathbf{0}}\mathbf{,}{\mathit{r}}_{\mathbf{1}}\mathbf{,}{\mathit{r}}_{\mathbf{2}}\mathbf{,}{\mathit{r}}_{\mathbf{3}}\mathbf{\}}$is a normal subgroup ofby listing all its right and left cosets.

Short answer

It is proved that$\mathit{N}\mathbf{=}\mathbf{\{}{\mathit{r}}_{\mathbf{0}}\mathbf{,}{\mathit{r}}_{\mathbf{1}}\mathbf{,}{\mathit{r}}_{\mathbf{2}}\mathbf{,}{\mathit{r}}_{\mathbf{3}}\mathbf{\}}$is a normal subgroup of${\mathit{D}}_{\mathbf{4}}$.

Step-by-step solution

Find the left cosets ofN

Form the multiplication table of${\mathit{D}}_{\mathbf{4}}$as :

$\mathbf{.}$	${\mathit{r}}_{\mathbf{0}}$	${\mathit{r}}_1$	${\mathit{r}}_{\mathbf{2}}$	${\mathit{r}}_{\mathbf{3}}$	$\mathit{d}$	$\mathit{h}$	$\mathit{t}$	$\mathit{v}$
localid="1657366639324" ${\mathit{r}}_{\mathbf{0}}$	localid="1657366834319" ${\mathit{r}}_{\mathbf{0}}$	${\mathit{r}}_1$	${\mathit{r}}_{\mathbf{2}}$	${\mathit{r}}_3$	$\mathit{d}$	$\mathit{h}$	$\mathit{t}$	$\mathit{v}$
localid="1657366483776" ${\mathit{r}}_1$	localid="1657366887273" ${\mathit{r}}_1$	role="math" localid="1657367231417" ${\mathit{r}}_{\mathbf{2}}$	${\mathit{r}}_3$	${\mathit{r}}_0$	$\mathit{h}$	$\mathit{t}$	$\mathit{v}$	$\mathit{d}$
localid="1657367171754" role="math" ${\mathit{r}}_{\mathbf{2}}$	${\mathit{r}}_{\mathbf{2}}$	$\mathit{r}\mathbf{3}$	${\mathit{r}}_0$	${\mathit{r}}_1$	$\mathit{t}$	$\mathit{v}$	$\mathit{d}$	$\mathit{h}$
localid="1657366545122" ${\mathit{r}}_{\mathbf{3}}$	${\mathit{r}}_{\mathbf{3}}$	${\mathit{r}}_{\mathbf{0}}$	${\mathit{r}}_1$	${\mathit{r}}_{\mathbf{2}}$	$\mathit{v}$	$\mathit{d}$	$\mathit{h}$	$\mathit{t}$
localid="1657366702264" $\mathit{d}$	localid="1657366730394" $\mathit{d}$	$\mathit{v}$	$\mathit{t}$	$\mathit{h}$	${\mathit{r}}_{\mathbf{0}}$	${\mathit{r}}_3$	${\mathit{r}}_{\mathbf{2}}$	role="math" localid="1657367388720" ${\mathit{r}}_{\mathbf{1}}$
$\mathit{h}$	$\mathit{h}$	$\mathit{d}$	$\mathit{v}$	$\mathit{t}$	${\mathit{r}}_1$	${\mathit{r}}_{\mathbf{0}}$	${\mathit{r}}_3$	${\mathit{r}}_{\mathbf{2}}$
localid="1657366862772" $\mathit{t}$	localid="1657367880288" $\mathit{t}$	$\mathit{h}$	$\mathit{d}$	$\mathit{v}$	${\mathit{r}}_{\mathbf{2}}$	${\mathit{r}}_1$	${\mathit{r}}_{\mathbf{0}}$	role="math" localid="1657367354040" ${\mathit{r}}_{\mathbf{3}}$
$\mathit{v}$	localid="1657367887548" $\mathit{v}$	$\mathit{t}$	$\mathit{h}$	$\mathit{d}$	${\mathit{r}}_{\mathbf{3}}$	${\mathit{r}}_{\mathbf{2}}$	${\mathit{r}}_{\mathbf{1}}$	${\mathit{r}}_{\mathbf{0}}$

Therefore, the left cosets of $\mathit{N}$are$\mathit{N}\mathbf{,}\mathit{d}\mathit{N}\mathbf{,}\mathit{h}\mathit{N}\mathbf{,}\mathit{t}\mathit{N}\mathbf{}$and$\mathit{v}\mathit{N}$.This implies that:

localid="1657366024467" $\mathit{N}\mathbf{=}\mathbf{\{}{\mathbf{r}}_{\mathbf{0}}\mathbf{,}{\mathbf{r}}_{\mathbf{1}}\mathbf{,}{\mathbf{r}}_{\mathbf{2}}\mathbf{,}{\mathbf{r}}_{\mathbf{3}}\mathbf{\}}$

localid="1657366043429" $\mathit{d}\mathit{N}\mathbf{=}\mathbf{\{}\mathit{d}\mathbf{,}\mathit{v}\mathbf{,}\mathit{t}\mathbf{,}\mathit{h}\mathbf{\}}$

localid="1657366099069" $\mathit{h}\mathit{N}\mathbf{=}\mathbf{\{}\mathit{h}\mathbf{,}\mathit{d}\mathbf{,}\mathit{v}\mathbf{,}\mathit{t}\mathbf{\}}$

localid="1657366123014" $\mathit{t}\mathit{N}\mathbf{=}\mathbf{\{}\mathit{t}\mathbf{,}\mathit{h}\mathbf{,}\mathit{d}\mathbf{,}\mathit{v}\mathbf{\}}$

localid="1657366155656" $\mathit{v}\mathit{N}\mathbf{=}\mathbf{\{}\mathit{v}\mathbf{,}\mathit{t}\mathbf{,}\mathit{h}\mathbf{,}\mathit{d}\mathbf{\}}$

Find the right cosets of N

It is observed that $\mathit{d}\mathit{N}\mathbf{=}\mathit{h}\mathit{N}\mathbf{=}\mathit{t}\mathit{N}\mathbf{=}\mathit{v}\mathit{N}\mathbf{=}\mathbf{\{}\mathit{v}\mathbf{,}\mathit{t}\mathbf{,}\mathit{h}\mathbf{,}\mathit{d}\mathbf{\}}$,and the right cosets are:

$\mathit{N}\mathbf{=}\mathbf{\{}{\mathbf{r}}_{\mathbf{0}}\mathbf{,}{\mathbf{r}}_{\mathbf{1}}\mathbf{,}{\mathbf{r}}_{\mathbf{2}}\mathbf{,}{\mathbf{r}}_{\mathbf{3}}\mathbf{\}}$

$\mathit{N}\mathit{d}\mathbf{=}\mathbf{\{}\mathit{d}\mathbf{,}\mathit{v}\mathbf{,}\mathit{t}\mathbf{,}\mathit{h}\mathbf{\}}$

$\mathit{N}\mathit{h}\mathbf{=}\mathbf{\{}\mathit{h}\mathbf{,}\mathit{d}\mathbf{,}\mathit{v}\mathbf{,}\mathit{t}\mathbf{\}}$

$\mathit{N}\mathit{t}\mathbf{=}\mathbf{\{}\mathit{t}\mathbf{,}\mathit{h}\mathbf{,}\mathit{d}\mathbf{,}\mathit{v}\mathbf{\}}$

$\mathit{N}\mathit{v}\mathbf{=}\mathbf{\{}\mathit{v}\mathbf{,}\mathit{t}\mathbf{,}\mathit{h}\mathbf{,}\mathit{d}\mathbf{\}}$

This implies that$\mathit{d}\mathbf{=}\mathit{N}\mathit{h}\mathbf{=}\mathit{N}\mathit{t}\mathbf{=}\mathit{N}\mathit{v}\mathbf{=}\mathbf{\{}\mathit{v}\mathbf{,}\mathit{t}\mathbf{,}\mathit{h}\mathbf{,}\mathit{d}\mathbf{\}}$.Therefore,$\mathit{d}\mathit{N}\mathbf{=}\mathit{h}\mathit{N}\mathbf{=}\mathit{t}\mathit{N}\mathbf{=}\mathit{v}\mathit{N}\mathbf{=}\mathit{N}\mathit{d}\mathbf{=}\mathit{N}\mathit{h}\mathbf{=}\mathit{N}\mathit{t}\mathbf{=}\mathit{N}\mathit{v}$.This implies that is the normal subgroup.