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 $f: \mathbb{Z} \rightarrow D_{5}$.



(ii) Determine the number of group homomorphisms $g: \mathbb{Z}_{7} \rightarrow D_{7}$



Hint: What can you can about the order of $f(x)$ where $x$ is an element of $G$? Also, for (ii) you might want to make use of the relation $(\ast)$.