Homomorphism from $S_3$ to ($\mathbb{Q},+)$ I am solving exercise in abstract algebra and could not solve this 1 correctly.

Does there exists a homomorphism from $S_3$ to the additive group ($\mathbb{Q},+)$ of rational numbers?

I think it exists. Map $A_3$ to $1$ and remaining elements to $-1$. But answer is no.
So, what mistake I am making? Please tell.

$\{0\}$ is the only finite subgroup of $(\Bbb Q,+)$, and hence there is no nontrivial homomorphism from any finite group $G$ to $(\Bbb Q,+)$.
