G=S_3 (symmetry group), H=\langle(123) \rangle , K=$\langle$(12)$\rangle$. Why G is said not isomorphic to external direct product of H and K?

Question

Let $\phi $ be function from G to H $\oplus$ K such that $\phi$ (123)=((123),(1)), $\phi$ (132)=((13)(1)) and etc. I has checking some case like $\phi$((123)(132))=$\phi$(123).$\phi$(132) and etc. Its look that there are isomorphism from G to H $\oplus$ K, but the book said the otherwise. Where is my fault?

Answer 1

Your proposed $\psi$ is not a homomorphism, since the order of $(13)$ in $S_3$ is $2$, but $\psi((13))$ must be either $((123), (12))$ or $((132), (12))$, which both have an order of $6$ in $\langle (123) \rangle \oplus \langle (12) \rangle$.