My textbook says:
Let $G$ act transitively on a set $X$. Fix $x_0 \in X$, let $H$ = Stab($x_0)$, and let $Y$ denote the set of conjugates of $H$ in $G$. Show that there is a $G$-equivariant surjective map from $X$ to $Y$ given by $x \to \text{Stab}(x)$.
So if we let $G$ be $S_3$ and $X = \{1, 2, 3\}$ and $f(x) = \text{Stab}(x)$, then $f(3) = \{e, (1 2)\}$, where $e$ is the identity permutation and $(1 2)$ is the two-cycle interchanging 1 and 2. Then if the claim up there is true then $\text{Stab}(1) = (1 3)\{e, (1 2)\} = \{(1 3), (1 3)(1 2)\}$ which is patently false. Where did I go wrong? Thanks!