I have a doubt in Dummit's Abstract Algebra on page115 :
Let $A = \{ 1 , 2, . . . , n \}$ , let $\sigma$ be an element of $S_n$ and let $G=\langle\sigma\rangle$ . Then $\langle\sigma\rangle$ acts on $A$ and so, by Proposition 2, it partitions $\{ 1 , 2, . . . , n\}$ into a unique set of (disjoint) orbits. Let $\mathcal{O}$ be one of these orbits and let $x\in\mathcal{O}$ . By (the proof of) Proposition 2 applied to ${\color{red}{A = \mathcal{O}}}$ we see that there is a bijection between the left cosets of $G_x$ in $G$ and the elements of $\mathcal{O}$ , given explicitly by $$ \sigma^ix\mapsto\sigma^iG_x\ . $$
My doubt is about the red part "${\color{red}{A = \mathcal{O}}}$", I think it is not certainty true:
Suppose $\mathcal{O}_x=A$ then for all $a\in A$ there must exist some $i\in A$ s.t. $\sigma^i\cdot x=\sigma^i(x)=a$ .But since $\sigma$ is arbitrary, it may not always hold.
For instance, let $\sigma=(1\ 2)\in S_3$ and $x=1$ then $\mathcal{O}_x=\{1,2\}\ne\{1,2,3\}=A$ .
So, is it a typo?