There are $n$ homomorphisms from the group $Z/nZ$ to the additive group of rationals $Q$

Question

There are $n$ homomorphisms from the group $Z/nZ$ to the additive group of rationals $Q$.

how can i find that the above statement is true/false

Answer 1

Hint: Elements of $\mathbb{Q}$ apart from the identity have infinite order, and all elements of $\mathbb{Z}/n\mathbb{Z}$ have finite order. Can a group homomorphism map an element of finite order to an element which does not have finite order?

Answer 2

Say the image of $1\in Z_n$ is $x\in Q$ then $nx=0$ in the rationals so $x=0$. Thus all homomorphisms are trivial, all elements are sent to $0$.