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Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$.

Show that either $|X| \le 2$ or $|X| \ge n$.

Small steps towards the solution:

  • As $S_n$ acts transitively on a set $X$, the whole of $X$ is one single orbit under the action of $S_n$.

  • By the Orbit-Stabilizer Theorem, then, $|X|$ = $|S_n : \text{Stabilizer of }x|$ for any $x \in X$.

  • We also know that the Stabilizer of any $x \in X$ is a subgroup of $S_n$.

  • When $|X| = 2$, the Stabilizer of $x$ is the alternating group $A_n$.

I'm halfway but can't get the final result. Any help would be much appreciated, as always. Thank you.

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    $\begingroup$ You can use $\TeX$ on this site. To do so, enclose it in dollar signs; single dollar signs for inline formulas, double dollar signs for displayed equations. You can also see the $\TeX$ code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands". $\endgroup$
    – joriki
    Aug 11, 2012 at 9:46
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    $\begingroup$ Following joriki's comment, I've $\LaTeX$'d the post; click 'edit' to see what I did to get the subscripts, $\le$ signs, etc. $\endgroup$ Aug 11, 2012 at 10:36
  • $\begingroup$ Thank you so much joriki & Clive -- this helps me out a lot. I'm not that familiar with typesets yet. $\endgroup$
    – Conan Wong
    Aug 11, 2012 at 14:13

1 Answer 1

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For $n < 4$ the result is clear. For $n = 4$ the result is false - we have a surjection $S_4 \to S_3$ by killing the unique normal subgroup of order 4, given by the products of transpositions (thanks to cocopuffs for fixing an error in this before).

For $n > 4$, the map $S_n \to \text{Aut}_\text{Set}(X) = S_{\# X}$ induced by your action must either be injective or have kernel $A_n$ or $S_n$. In the first case we must have $\#X \ge n$ and in the others we have $\#X = 2$ or $1$ respectively.

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    $\begingroup$ But the coset space has $6$ elements in the case $n=4$ $\endgroup$
    – Cocopuffs
    Aug 11, 2012 at 9:57
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    $\begingroup$ Yes. Maybe you mean let $S_{4}$ act by conjugation on the $3$ non-identity elements of that normal Klein $4$-subgroup. That is a transitive action of $S_{4}$ on a set $X$ of size $3.$ $\endgroup$ Aug 11, 2012 at 10:00
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    $\begingroup$ @Geoff Robinson that's better (more concerete) than what i meant, i meant that the quotient by that group is $S_3$ so we get a transitive action on a 3 element set. $\endgroup$
    – user29743
    Aug 11, 2012 at 10:03
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    $\begingroup$ @Cocopuffs the question is equivalent to the existence of an arbitrary subgroup of index between $2$ and $n$, not just a normal one. $\endgroup$
    – user29743
    Aug 11, 2012 at 10:04
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    $\begingroup$ @Cocopuffs: Your first comment about the index was right. Your second comment isn't correct (you dohave the right nomal subgroups, of course). $S_{4}$ does have a subgroup of index $3$, a Sylow $2$-subgroup. Therefore, there is a homomorphism from $S_{4}$ to $S_{3}$ (in fact surjective). The kernel is the normal Klein $4$-subgroup of course. $\endgroup$ Aug 11, 2012 at 10:06

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