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 ?
Subgroups between $S_n$ and $S_{n+1}$
12
$\begingroup$
group-theory
permutations
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1What can you say about an element of $H$ which is not in $S_n$? – 2012-07-19
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0Stabilizers in primitive (hence 2-transitive) groups are maximal. – 2012-07-19
1 Answers
14
None.
Let $S_n
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0Sorry I used a non-standard notation for cycles (the commas aren't usually there) but I couldn't find a way to manage spaces in this TeX environment. – 2012-07-19
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4If by none you mean two... (the inclusions are not necessarily proper on either end!) – 2012-07-19
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1I often put in ~'s for cycle notation. e.g. `$(1~2~3)$`=$(1~2~3)$ – 2012-07-19
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1@QiaochuYuan: of course you are right. The "none" incipit was chosen for dramatic effects :) I guess that I should say that I was impliciting assuming that $H$ is not one of the trivial possibilities. – 2012-07-19
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0@anon: thanks for the tip. – 2012-07-19