The dihedral group $D_{5}$ is generated by an element $a$ of order $5$, and an element $b$ of order 2, satisfying the relation

$(\ast)$ $ba = a^{5-1}b$

(i) Determine number of group homomorphisms $f: D_{5} \rightarrow {\mathbb{Z}}$.



(ii) Determine the number of group homomorphisms $g: D_{7} \rightarrow {\mathbb{Q}}^\times$



Hint:
What can you can about the order of $f(x)$ vs. the order of $x$?
What can you say about the order of elements of $\mathbb{Z}$ and of $\mathbb{Q}^\times$?
And for (ii): you might want to make use of the relation $(\ast)$.