Let $S_3$ act on the set $A = \{ (i,j) \ |\ 1 \leq i,j \leq 3 \}$ by $\sigma \cdot (i,j) = (\sigma(i), \sigma(j))$.
Find the orbits of $S_3$ on $A$.
So I know I have to find a set $\{\sigma \cdot (i,j) | \sigma \in S_3 \}$ for each $(i,j)$. But I don't understand how $S_3$ acts on $A$. Up until now we always got some expression for $\sigma$ like $\sigma = (123)$.
If we take $(2,3)$ for example, how would $S_3$ act on $(2,3)$? $\sigma \cdot (2,3) = (\sigma(2), \sigma(3))$, but how do I determine what $\sigma$ is?
$\sigma\cdot(2,3)=\sigma(2),\sigma(3))$ holds for all $\sigma\in S_3$. So for example if $\sigma=(1\,2\,3)$, then $\sigma(2)=3$ and $\sigma(3)=1$, so $\sigma\cdot(2,3)=(3,1)$.