Slowly increasing smooth mappings with values in a Lie group?

Question

Let $G$ be $l$-dimensional compact Lie group and consider any smooth $F : \mathbb{R}^n \to G$.

Then, the first-order derivative of $F$ at each $x \in \mathbb{R}^n$ can be regarded as a linear mapping $D_x F : \mathbb{R}^n \to \mathbb{R}^l$. Let us fix a matrix norm $\lVert \cdot \rVert$ and assume that \begin{equation} x \mapsto \lVert D_x F \rVert \text{ is polynomially bounded} \end{equation} We can extend the above notion of polynomial boundedness to higher order derivatives as well.

Now, let us define a set $\mathfrak{G}$ as the set of $F : \mathbb{R}^n \to G$ such that $F$ is smooth and all its derivatives are polynomially bounded in the above sense.

Now, my question is that

Is this $\mathfrak{G}$ a group under pointwise multiplication and inverse?

That is,

1. For any $F_1, F_2 \in \mathfrak{G}$ and $x \in \mathbb{R}^n$, let $(F_1 F_2)(x) := F_1(x) F_2(x)$. Then, do we have $F_1 F_2 \in \mathfrak{G}$?

2. For any $F \in \mathfrak{G}$ and $x \in \mathbb{R}^n$, let $F^{-1}(x) := [F(x)]^{-1}$. Then, do we have $F^{-1} \in \mathfrak{G}$.

I think the item 1 must be true, but not sure about item 2. For example, can a derivative of $F$ decay exponentially at infinity, making a derivative of $F^{-1}$ "grow exponentially" at infinity?

Could anyone please clarify for me? If I am wrong about $\mathfrak{G}$ being a group, what additional conditions should I impose to make it into a group?

Add) I think this is indeed the case for $\operatorname{SU}(N)$ with $N \geq 2$ or any direct product of them. Still, I haven't figured out for general $G$.

Add 2) This ME post shows that my guess is also true for $\operatorname{U}(1)$.

Answer 1

Note that the maps $$\iota: G\to G,\ x\mapsto x^{-1}\qquad \mu:G\times G\to G,\ (x,y)\mapsto x\cdot y$$ have bounded derivatives, since you are assuming $G$ to be compact.

Then for $F_1, F_2\in\mathfrak G$ you have that the derivatives of $$F_1^{-1}=\iota\circ F_1,\qquad F_1\cdot F_2 = \mu\circ (F_1,F_2)$$ will have polynomial growth, being the composition of a map whose derivative has polynomial growth and a map with bounded derivative.