Definition of $\Gamma$-invariant set with $\Gamma \subset O(n)$

Question

I am reading a paper, the first sentence of "Preliminaries and notations" says

We assume $\Gamma$ to be a finite subgroup of $O(n)$ and $X\subset \mathbb{R}^n$ to be a compact $\Gamma$-invariant set.

I am a bit confused about the general used definition of $\Gamma$-invariant set here will be (that paper does not give the definition for it).

My understanding is the following:

$X$ is called a $\Gamma$-invariant set, if for each $x\in X$, $$\{\gamma x \mid \gamma\in \Gamma\}\subset X,$$i.e., the orbit of $x$ under each $\gamma \in \Gamma$ is in $X$ and this should hold for all $x\in X$.

My questions are

1. Is the above definition correct?
2. Does "$\Gamma$-invariant set $X$" imply $\gamma X=X$ and $\gamma^{-1} X=X$?
3. If not, which conditions should be added to make 2. correct?

thanks!

Answer 1

The definition says that for all $\gamma$, $\gamma X \subset X$. This includes the case $\gamma^{-1}X$ as $\Gamma$ is a subgroup so $\gamma^{-1} \in \Gamma$. To see it is the whole set note $x = \gamma.(\gamma^{-1}x)$, where $\gamma^{-1}x \in X$ as we have already noted.