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Show that the group $S_n$ (right) acts on the set of subsets of $\{1,2,\ldots n\}$ by $\rho(S,\sigma)=S\sigma$.

Show that there are $n+1$ orbits, one for each possible value of $|S|$. Show that if $|S|=k$ then $\mathrm{Stab}(S)\cong S_k\times S_{n-k}$.

I have shown that there are n+1 orbits but now I am stuck with the last bit.

my initial thought is that if I let $S =\{1,2,3,...,k\}$ then $\mathrm{Stab}(S)$ would involve $(1)(2)(3)...(k)$ i.e. mapping to itself and then $k+1$ to $n$ mapped in any ways ($S_{n-k}$)

I feel like $\mathrm{Stab}$ is isomorphic to $S_{n-k}$ what am I misunderstanding here? How would I prove the last part?

Thanks a lot

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You're missing that $\sigma$ need not fix $S$ pointwise, but rather only setwise. That's where you get the initial $S_k$ factor.

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  • $\begingroup$ Aha! Thanks for your reply $\endgroup$ – user120227 May 13 '14 at 23:02

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