6
$\begingroup$

Let $G$ be a transitive permutation group on a set $X$ with $n$ elements. Assume that $G$ is primitive, i.e., $G$ preserves no non-trivial partition of $X$. Assume as well that $G$ is not $2$-transitive, i.e., the action of $G$ on pairs of distinct elements of $X$ is not transitive. Because $G$ is not $2$-transitive, the action of a pointwise stabilizer $G_x$ ($x\in X$) on $X\setminus \{x\}$ cannot be transitive.

Question: how large can the largest orbit of $G_x$ be? Is it possible that it be of size $\geq 0.99n$, say?

(If $G$ is not required to be primitive, then $G_x$ can have very large orbits. For example, for $n$ even, let $G$ be generated by transpositions $(2m-1\;\; 2m)$ and by elements of $\text{Sym}(n/2)$ acting on the $n/2$ possible pairs $\{2m-1,2m\}$ of elements of $\{1,2,\dotsc,n\}$. Then $G$ is not $2$-transitive, but $G_x$ has $\{3,4,\dotsc,n\}$ as an orbit.)

$\endgroup$
  • $\begingroup$ had a hard time understanding your example, but I see now. $x=1$ and you're showing the orbit of 3 under the stabilizer of $x$. $\endgroup$ – Anthony Quas Mar 29 '14 at 19:06
12
$\begingroup$

Let $G=S_n$, acting on the ${{n} \choose {2}}$ $2$-sets from $[n]$. The stabilizer of a $2$-set $X$ is maximal in $S_n$ and has two orbits on the remaining $2$-sets, one consisting of those that intersect $X$ nontrivially. The other orbit has size ${{n-2} \choose {2}}$. Now let $n \rightarrow \infty$.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.