Prove that $H\times K \cong K\times H$

Question

According to the book:

Let $G$ be the internal direct product of subgroups $H$ and $K$. Then $G$ is isomorphic to $H\times K$.

From that it results $H\times K \cong K\times H$. Is there any 'direct' proof for $H\times K \cong K\times H$ i.e. without using the mentioned theorem?

Answer 1

There's an extremely obvious bijective homomorphism to use between the two, namely

$$\phi(h,k)=(k,h)$$

Also I'll add that you may not know what the significance of a bijective homomorphism is, or what a homomorphism even is. If it'll help you for me to expand on the terms I've used, just leave a comment letting me know and I'll give a more thorough explanation.