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

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

2 Answers 2

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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?

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