How many group homomorphisms from $\mathbb{Q/Z\to Q}$ are there?

Question

I was asked a question in RKMVERI interview for MSc Mathematics.How many group homomorphisms are there from $\mathbb{Q/Z}$ to $\mathbb Q$?I think there is only one,the trivial one,but I am unable to prove it.Can someone help?

Answer 1

Let $f : \mathbb{Q/Z\to Q}$ be a group homomorphism.

Take any $a = m/n \in \mathbb Q/\mathbb Z$. Summing $n$ copies of $a$ will give zero in the ring $\mathbb{Q/Z}$.

$\sum_{i=1}^n a = 0$. Therefore $0 = f(\sum_{i=1}^n a) = \sum_{i=1}^n f(a) = n \cdot f(a)$.

But this cannot hold for any element of $\mathbb Q$ other than $0$. This implies $f(a) = 0$ for all $a$.