Homomorphism from $S_3$ to ($\mathbb{Q},+)$

Question

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.

Answer 1

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

Answer 2

Your map is a homomorphism to the multiplicative group of non-zero rational numbers.

You could map every element of $S_3$ to $0$ in $\mathbb Q$

to obtain a homomorphism to the additive group of rational numbers.