Exercise 2.1.3 (Some subgroups of $D_8$)

Proof. We know that $D_8=⟨r,s⟩$, where $r^4=s^2=e$ and $\mathit{rs}=s{r}^{-1}$.

(a)

Using $s{r}^2=r^{2}s$, we obtain

$\text{⋅}$	$1$	$r^2$	$s$	$s{r}^2$
$1$	$1$	$r^2$	$s$	$s{r}^2$
$r^2$	$r^2$	$1$	$s{r}^2$	$s$
$s$	$s$	$s{r}^2$	$1$	$r^2$
$s{r}^2$	$s{r}^2$	$s$	$r^2$	$1$

Therefore $H=\{1,r^2,s,s{r}^2\}=⟨r^2,s⟩\ne \varnothing$ is stable by product and inverses( $x^{-1}=x$ for all $x\in H$), so $H$ is a subgroup of $D_8$.

(b)

Similarly, using $\mathit{rs}=s{r}^3$, we obtain

$\text{⋅}$	$1$	$r^2$	$\mathit{sr}$	$s{r}^3$
$1$	$1$	$r^2$	$\mathit{sr}$	$s{r}^3$
$r^2$	$r^2$	$1$	$s{r}^3$	$\mathit{sr}$
$\mathit{sr}$	$\mathit{sr}$	$s{r}^3$	$1$	$r^2$
$s{r}^3$	$s{r}^3$	$\mathit{sr}$	$r^2$	$1$

So $K=\{1,r^2,\mathit{sr},s{r}^3\}$ is a subgroup of $D_8$.

Note that $H\simeq K\simeq {\mathbb{Z}}_2\times {\mathbb{Z}}_2$. □