Exercise 2.3.11 (Cyclic subgroups of $D_8$)

Proof. We know that

(You may prefer $D_8=\{e,r,r^2,r^3,s,\mathit{sr},s{r}^2,s{r}^3\}$.) The order of $r$ is $4$ (thus $r^2$ has order $2$ ,and $r,r^3$ have order $4$), and $s,\mathit{rs},r^{2}s,r^{3}s$ have order $2$. We obtain all the cyclic groups of $D_8$:

• order 4: $⟨r⟩$ ( $=⟨r^3⟩$),
• order 2: $⟨s⟩,⟨\mathit{rs}⟩,⟨r^{2}s⟩,⟨r^{3}s⟩$ and $⟨r^2⟩$,
• order 1: $⟨e⟩$

The subgroup $⟨r^2,s⟩=\{e,r^2,s,r^{2}s\}$ is not cyclic, because all of its elements have order $1$ or $2$, so there is no element of order $4$ in $⟨r^2,s⟩$.

(Another such subgroup is $⟨r^2,\mathit{sr}⟩=\{e,r^2,\mathit{sr},s{r}^3\}$: see Exercise 2.1.3.) □