There is a exercise problem that asks "Identify a set with another set ". I don't understand what I should do. Do I need to establish a bijection between them? Thanks
EDIT-I: Actual question: G is a group. S is a set. G acts on S. For $x\in S$, $G.s:=\{g.s|g\in G\}$ is a orbit. $G_s:=\{g\in G|g(s)=s\}$ is an isotropy group. Now I need to identify $G.s$ with the quotient $G/G_s:=\{gG_s|g\in G\}$ by the action of $G_s$ on the right.
I found a following bijection: $$\alpha: G/G_s \to G.s, \ gG_s \mapsto g.s$$
Is this enough?
The title doesn't fit to the question. $G/G_s$ is not a group. It is a left $G$-set. Also the orbit $Gs$ is a left $G$-set. And the actual statement is that these two left $G$-sets are isomorphic. Your isomorphism is correct.