Exercise 3.5.6 ($\langle (1\ 3), (1\ 2 \ 3 \ 4) \rangle \simeq D_8$)

Proof. Put $\sigma =(1234)$ and $\tau =(13)$. Then

Since $D_8=⟨r,s\mid r^4=s^2=1,\mathit{𝑟𝑠}=s{r}^{-1}⟩$, there is a surjective homomorphism $\phi :D_8\to ⟨\sigma ,\tau ⟩$ such that $\phi (r)=\sigma$ and $\phi (s)=\tau$. Therefore $|⟨\sigma ,\tau ⟩|\le |D_8|=8$. Moreover,

where these $8$ permutations are distinct, so $|⟨\sigma ,\tau ⟩|\ge 8$. This shows that $|⟨\sigma ,\tau ⟩|=8$, therefore $\phi$ is an isomorphism, so

(Alternatively, If we choose the numbering $1,2,3,4$ to the vertices of a square as in the figure p. 24, then $\sigma$ corresponds to the rotation of angle $\pi ∕2$, and $\tau$ to the reflection about the line $(2,4)$. These two geometric transformations are generators of $D_8$.) □