The question I want to ask is different from this one.
Let $M$ be a smooth manifold which admits a group structure such that the multiplication map $m:G\times G\to G$ defined as $m(g, h)=gh$ for all $g, h\in G$ is a smooth map. Then $G$ is a Lie group.
Hint from professor Lee's book: The map $F\colon G\times G\to G\times G$ defined by $F(g,h) = (g,gh)$ is is a bijective local diffeomorphism.
By taking advantage of the result here(this is the problem 7-2 of Lee's book), I can prove that $dF_{(e,e)}$ is an isomorphism from $T_eG\oplus T_eG$ to itself sending $(X,Y)$ to $(X,X+Y)$, which implies that $F$ is a local diffeomorphism at $(e,e)$. But I don't know how to use this result to obtain that $F$ is local diffeomorphism everywhere. And I also find it hard to prove that $dF_{(g,h)}$ is an isomorphism for general element $(g,h) \in G\times G$.
Thanks in advance!