Homomorphism between groups and group order

Question

Let $φ: G \to \mathbb{Z_{15}}$ be a group Homomorphism and $ord(G) = n$.

Are the following true or false?

1. 15|n
2. n|15

All I know is that for $a\in G, ord(φ(a))|ord(a)$, but I can't seem to be getting anywhere from here. Any tips?

Answer 1

With no additional assumption, there is no way to deduce any kind of equation (you can check my comment above). However, if $\varphi$ is surjective, you can take $g\in G$ with $\varphi(g)=1$ and then deduce that order of $g$ is divisible by $15$ which implies that the order of $G$ is divisible by $15$.