Simple expression for the joint orbit of two permutations
Solution 1:
This isn't an equivalence relation. In your example, $1 \sim 2$ and $2 \sim 3$, but $1 \nsim 3$.
In your revised question, the easiest result is that the size of the orbit of $j$ is the index of its stabilizer: $\vert \mathscr O(j) \vert = [H:C_H(j)]$, where $C_H(j)= \{ \rho \in H \mid \rho(j)=j \}$. That means that the size of each orbit must divide $\vert H \vert$, which may help determine how many orbits there can be.