I want to prove that the direct product $G\times H$ of two groups has a subgroup isomorphic to $G$ and a subgroup isomorphic to $H$.
How I thought to prove is that taking a pair $(g,h)$ from $G\times H$ and then to show that for some $g$ in $G$ there is an isomorphism with the pair $(g,h)$ of $G\times H$. and the same for $h$ in $H$ also. Is this method correct? Or should I try to show that there is a bijection between the groups and there exists an equivalence relation between the groups?
Please explain how should I proceed.
What do you think about these subgroups
$$K_1=\{(g,e): g\in G\}$$ $$K_2=\{(e,h): h\in H\}$$
rspectively isomorphic to $G$ and $H$?