Problem 7.1 of Lee's Introduction to Smooth Manifolds (2nd Edition) reads:
Show that for a Lie group $G$, the multiplication map $\mu:G\times G\to G$ is a submersion (Hint: Use Local Sections).
I did the following:
Fix $g, h\in G$. Then since $T_{(g, h)}(G\times G)\cong T_g G\oplus T_h G$, we have
$$ d\mu_{(g, h)}(X, Y)= d(\mu\circ i^h)_gX+d(\mu\circ j^g)_h Y $$ for $X\in T_gG, Y\in T_hG$, where $i^h:G\to G\times G$ is the map defined as $i^h(x)=(x, h)$ for all $x\in G$ and similarly for $y^g$.
Thus we have
$$ d\mu_{(g, h)}(X, Y)= dR_h|_gX+dL_g|_hY $$
Since $dR_h|_g:T_gG\to T_{gh}G$ is a linear isomorphism, we see that the rank of $\mu$ is full. So we are done.
I do not see how to do it using the hint Lee has given
Can somebody please do it using the hint?