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This is the problem 7-6 of Lee's Introduction to Smooth Manifolds (2nd edition):

Suppose G is a Lie group and U is any neighborhood of the identity. Show that there exists a neighborhood V of the identity such that $V \subset U$ and $gh^{-1} \in U$ whenever $g, h \in V$.

How do I approach this problem? I tried using the smoothness of $(g,h) \mapsto gh^{-1}$ or the open subgroup generated by $U$, but it didn't get me anywhere.

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You don’t need anything fancy here. Just use continuity and pick $V$ carefully.

The map $f : G \times G \to G$ defined as $f(g,h) = gh^{-1}$ is a smooth map. Suppose $U$ is a neighbourhood of the identity $e\in G$. By continuity, $W = f^{-1}(U)$ is open in $G \times G$. Since $(e,e) \in W$, there are neighbourhoods $W_1,W_2 \subseteq G$ containing $e\in G$ such that $W_1\times W_2 \subseteq W$. Choose $V$ as

$$V = (W_1 \cap W_2) \cap U$$

So $V \times V \subseteq W=f^{-1}(U)$ implies $f(V \times V) \subseteq U$. I.e., $\forall g,h \in V$, we have $f(g,h)=gh^{-1} \in U$.

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