Let $G$ be a group with identity element $e$. Let $H$ and $K$ be normal subgroup of $G$ such that $H∩K={e}$. Prove that
$(H×K)≅(H⊕K)$
$(H×K)$ is the internal direct product between $H$ and $K$
$(H⊕K)$ is the external direct product between $H$ and $K$
I know that since H and K be normal subgroup of G such that $H∩K={e}$, $hk=kh$ for all $h∈H$ and $k∈K$.
I also know that $(h,k)∈(H⊕K)$ . But I’m not sure I know how to link them together and show that
$(H×K)≅(H⊕K)$
Best Answer
Hint. Show that every element of the internal direct product may be uniquely represented as a product $hk$ for some $h\in H$, $k\in K$. Then use this to show that $\varphi(hk)=(h,k)$ is an isomorphism.