Determine all the homomorphisms $\varphi:\Bbb Z_4 \to S_3$.
As $\Bbb Z_4$ is generated by $\langle 1 \rangle$ and since $\varphi(g^k)=\varphi(g)^k$ I have that $\varphi(1+1+1+1)=\varphi(0)=(1)$ since the homomorphism must map the identity to the identity.
It seems that I would need to figure out where $\varphi$ maps $1$. Since $\varphi(1+1+1+1)=\varphi(1)\varphi(1)\varphi(1)\varphi(1)=(1)^4$ but I'm a bit confused here. Should I find an element of $S_3$ for which multiplied by itself $4$ times I'll get the identity or what?