Find an example of a homomorphism of $D_4$ to $Z_8$ such that $\ker=\{e,R,F\}$.

Question

The question asks to find an example of a homomorphism of $D_4$ to $Z_8$ such that $\ker=\{e,R,F\}$.

$$D_4 = \{e, R,R^2,R^3,F,FR,FR^2,FR^3\} \text{ and } Z_8=\{0,\ldots,7\}$$

I thought about letting the map equal: $$\phi(e)=0$$ $$\phi(F)=0$$ $$\phi(R)=0$$

But I wouldn't know how to map the other elements $\phi(R^2),\phi(R^3),\ldots$

Answer 1

The kernel of a homomorphism must be a subgroup so this is impossible.