Exercise 3.2.13 ($D_8$ as a subgroup of $S_4$)

Proof. We use the numbering of the vertices of a square given p.24, and We define $D_8=⟨r,s⟩$, where $r$ is the rotation of angle $\pi ∕2$, and $s$ the symmetry with respect to the $x$-axis. We use the numbering of the vertices of a square given p.24. Then $D_8$ acts on the set $\{1,2,3,4\}$ by an action whose associate homomorphism $\phi :D_8\to S_4$ is defined by

Since an affine application of the plane is determined by three non-aligned points, this action is faithful, therefore

We identify $D_8$ with the subgroup $H\le S_4$ given by

Let $K=⟨(123)⟩\le S_4$. Since

the permutations $(1234)\in H$ and $(123)\in K$ do not commute in $S_4$.

(But since $H\cap K=\{1\}$, $|\mathit{𝐻𝐾}|=|H|\cdot |K|=8\cdot 3=24=|S_4|$, so $G=\mathit{𝐻𝐾}=\mathit{𝐾𝐻}$.) □