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$\begingroup$

Lets look at $S_n$ as subgroup of $S_{n+1}$. How many subgroups $H$, $S_{n} \subseteq H \subseteq S_{n+1}$ there are ?

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    Stabilizers in primitive (hence 2-transitive) groups are maximal.2012-07-19

1 Answers 1

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None.

Let $S_n and suppose that $H$ contains a cycle $c$ involving $n+1$, say $ c=(a,\cdots,b,n+1). $ Then by composing to the left with a suitable permutation $\sigma\in S_n such that $\sigma(a)=b$ we have $ \sigma c=\sigma^\prime (b,n+1) $ where $\sigma^\prime(n+1)=(n+1)$, i.e. $\sigma^\prime\in H$. Thus the transposition $(b,n+1)$ is in $H$. Of course we may assume that $b=1$ and so all transpositions $(1,2)$, $(1,3)$, ..., $(1,n+1)$ are in $H$. These transpositions are known to generate $S_{n+1}$.

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    @anon: thanks for the tip.2012-07-19