This is the final step into completing a problem and I am a bit stuck. I need to show that:
Consider the action of $S_{n-1}$ on $S_n/S_{n-1}$ by left multiplication. Does this action have exactly two orbits? IF not, is there an action that would give exactly two orbits?
Thank you in advance!
Hint: The left cosets of $\pi S_{n-1}$ with $\pi\in S_n$ can be characterized by $\pi(n)$, since all elements of $\pi S_{n-1}$ send $n$ to $\pi(n)$. Using this characterization, you can see which left cosets can be transformed into each other by left-multiplication with an element of $S_{n-1}$ and which can't.