In Serge Lang's Abstact ALgebra, he define orbits like this:
let G operate on a set S. Let $s \in S$. The subset of $S$ consisting of all elements $xs, x\in G$, is called the orbit of $s$ under $G$
How is this a subset of $S$? Can't there be an $x \in G$ such that $xs \not \in S$?
For example, if $S = e$, where $e$ is the identity element, then isn't the orbit of $e$ under $G$, $G$?
For each element $s\in S$, the orbit of $s$ is defined by $O_s = \{gs\mid g\in G\}$. Since $gs\in S$, this is clearly a subset of $S$.
In view of the identity element $e\in G$, by definition $es=s$ for each $s\in S$ and so the orbit of $s$ contains $s$ (choose $g=e$), i.e., $s\in O_s$.