@geocalc33 yes that is correct!
The stabilizer group of a point in the set $X$ on which the group $G$ acts.
It's a subgroup of $G$, called $\text{Stab}(x)$ for any given $x \in X$.
You should look up the Orbit-Stabilizer theorem
@geocalc33 for the soda can example though the set of rotations alone form a subgroup of $G$, $H$. Then $\text{Stab}_H(x) = \{ R_{2\pi \theta} : \theta \in \Bbb{Z}\}$.
where $R_{\theta}$ is rotation of the cylinder along its axis by $\theta$ radians
$\theta \in \Bbb{Z}$, not $\Bbb{R}$.
