Are the following definitions essentially the same:
Orbit:
Let $G$ be a group of permutations of a set $S$. For each $s \in S$, let $\operatorname{orb}_G(s)=
\{f(s) \mid f \in G\}$. The set $\operatorname{orb}_G(s)$ is a subset of $S$ called the orbit of $s$
under $G$
&
The orbit of a point $x \in X$ is the set of elements of $X$ to which $x$ can be moved by the elements of $G$. The orbit of $x$ is denoted by $Gx$:
$Gx = \{ g.x \mid g \in G\}$.
I guess that they are essentially the same definitions. How?
Let $X=S$, $x=s$ and $f=g$ and $\operatorname{orb} G(\cdot)=G\cdot$. Then the two definitions are the same.