Find the orbits of the set $X=\left \{ 1,2,3 \right \}$ under $S_{3}$.

Question

Find the orbits of the set $X=\left \{ 1,2,3 \right \}$ under $S_{3}$.

The answer is $O_{1}=O_{2}=O_{3}=\left \{ 1,2,3 \right \}$, but I do not understand why?

Answer 1

For each $x$ and each $y\in\{1,2,3\}$, there is a permutation $\rho\in S_3$ such that $\rho(x)=y$.

Such a group action is called transitive.

Answer 2

Any element can be taken to any other element by an element of $S_3$. For instance, $1$ maps to $2$ under the permutation $(1\:2)$, and $1$ maps to $3$ under $(1\:3)$.