$H$ and$K$ subgroups of $G$ Prove$\Bigl(H\cap K=\left\{1\right\}\land\left(h\in H,k\in K\Rightarrow hk=kh\right)\Bigr)\Rightarrow HK\approx H\times K$

Question

Let $H$and $K$ be normal subgroups of a group $G$. Prove that for $ \Bigl (H\cap K =\left\{1\right\}\land\left(h\in H ,k\in K\Rightarrow hk=kh\right)\Bigr)\Rightarrow HK\approx H\times K $

now I know that
$HK\le G\iff HK=KH$

for $HK=\left\{y\in G \mid y=hk, h\in H \land k\in K\right\}$

but I don't know how to take from here any ideas?

Answer 1

Hint: Show $\phi(h,k)=hk$ is an isomorphisms between $H\times K$ and $HK$. Doing it in that direction avoids having to show the map is well defined. Then just prove it is a homomorphism and bijective