Exercise 2.5.3 (The subgroup $\langle s, r^2 \rangle$ of $D_8$ is isomorphic to $V_4$)

Proof. First $⟨s,r^2⟩\supseteq \{1,s,r^2,s{r}^2\}$, where $r^2\in Z(D_8)$.

Moreover $\{1,s,r^2,s{r}^2\}$ is a subgroup by Exercise 2.1.3, therefore

The group $V_4$ has the same table that $V_4$, if we replace $a$ by $r^2$, $b$ by $s$ and $c$ by $s{r}^2$ (for instance), so $H\simeq V_4$.

Alternatively, all elements of $H$ have orders $1$ or $2$, thus $H$ is not cyclic. Since every group of order $4$ is isomorphic to $Z_4$ or $Z_2\times Z_2\simeq V_4$ (see Exercise 10), we obtain

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