Let $G$ be a group and let $θ: S_3 \rightarrow G$ be a homomorphism such that $θ((1 2)) = e_G$.
I know that $θ((12)$ is in the kernel, as it gives me the identity. The same could be said about $θ(1)$ because $θ((1))θ((12)) = θ((1)(12)) = θ(12) = e_G$.
However there are supposedly more elements from $S_3$ that are in the kernel as well, and I'm not sure how to find them. I've tried using the homomorphism property that $θ(x)θ(y) = θ(xy)$, but that's only helped me find that $θ(1)$ is in the kernel.