Problem 5.1.13 ( $Z_4 * D_8$ and $Z_4 * Q_8$)

Proof. We guess that

where $G$ is defined by generators and relations:

To prove (1), consider the elements

where $Z=\{(1,1),(x^2,r^2)\}$ as in Exercise 12.

Then

the last relation being true because

Hence there exists a unique homomorphism $\phi :G\to Z_4\ast D_8$ such that

Since $Z_4\ast D_8=⟨\tilde{x},\tilde{s},\tilde{r}⟩$, the homomorphism $\phi :G\to Z_4\ast D_8$ is surjective.

Since $\chi$ commutes with $\sigma$ and $\rho$, and since $\mathit{𝜌𝜎}=\sigma {\rho }^{-1}$, every element $\gamma \in G$ is of the form

and the relations ${\chi }^4={\rho }^4={\sigma }^2=1$ show that we can take

Then $\phi (\gamma )=\phi {(\chi )}^u\phi {(\sigma )}^v\phi {(\rho )}^w=(x^u,s^v{r}^w)Z$. Therefore

This shows

Then $\phi$ is injective, so $\phi$ is an isomorphism.

If we prefer, we can write by changing the name of the generators,

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