I would like some examples of groups $G$ satisfying all of the following criteria:

  1. $G<Sym(n)$, the symmetric group on $n$ letters, and $G$ is primitive.
  2. $G$ has a regular suborbit, i.e. if $M$ is a point-stabilizer, then there is an orbit of $M$ on which $M$ acts regularly.
  3. $G$ has a non-faithful suborbit, i.e. if $M$ is a point-stabilizer, then there is an orbit of $M$ on which $M$ acts non-faithfully.

Just to be clear: the suborbits in 2 and 3 need to be different! If possible, I would like to strengthen 3:

  1. $G$ has a non-faithful suborbit $\Lambda$ such that, if $M$ is the associated point-stabilizer, then $M^\Lambda\cong Alt(\Lambda)$ or $Sym(\Lambda)$. Here $M^\Lambda$ is the permutation group induced on $\Lambda$ by the action of $M$.

I don't know how common such groups are... So for now I'm just interested in collecting different examples to see if I can pick out some interesting properties. GAP/ MAGMA computations, as well as theoretical answers telling me what such groups might look like, are of interest to me.

Thanks in advance.

  • $\begingroup$ Since I'm not used to this terminology, let me say that regular means simply transitive. So 2 means that every point stabilizer $M$ acts freely on some $M$-orbit. $\endgroup$ – YCor Mar 7 '19 at 10:22
  • $\begingroup$ Your question doesn't make clear (to me) whether you already know any such example. $\endgroup$ – YCor Mar 7 '19 at 10:23
  • $\begingroup$ @YCor, No, I don't know any examples! Thanks for clarifying the terminology. $\endgroup$ – Nick Gill Mar 7 '19 at 11:09

It is easy to find examples that satisfy conditions 1 - 3. I searched through the Atlas of Finite Simple Groups, looking for simple groups with maximal subgroups that might provide examples, where we take $G$ to be the image of the action of the simple group $S$ on the cosets of the chosen maximal subgroup $H$, and I found lots, using Magma.

In particular, I found two examples satisfying 1 - 4, where the action of the point stabilizer on the non-faithful orbit is $S_3$. These were:

  1. $S={\rm PSL}(2,11)$, $H = D_{12}$.

  2. $S={\rm PSL}(2,13)$, $H = D_{12}$.

But I don't think there are any more examples of that type, and I suspect that there are not many examples altogether. There are papers classifying primitive permutation groups with suborbits of length at most $5$, which you could find by searching. For length $5$, this is done in a paper by Fawcett, Giudici, Li, and Praeger, and I checked that there are no examples.

Since you asked for details of computations, here is the Magma computation for the Example 1.

> G:=PSL(2,11);                            
> M:=[m`subgroup:m in MaximalSubgroups(G)];
> [#m: m in M];
[ 55, 12, 60, 60 ]
> P:=CosetImage(G,M[2]);
> S:=Stabiliser(P,1);             
> O:=Orbits(S);
> [#o: o in O];
[ 1, 3, 3, 6, 6, 6, 6, 12, 12 ]
> [#OrbitImage(S,o): o in O];
[ 1, 6, 6, 12, 12, 12, 12, 12, 12 ]
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  • $\begingroup$ Derek, thank you as always. What do you mean by your remark "I suspect that there are not many examples altogether"? Do you think that there may not be examples of groups satisfying (4) once $|\Lambda|$ is larger than some constant? If so, are there other "large" groups that could be induced on non-faithful orbits? By "large" I mean, growing exponentially with $|\Lambda|$, for instance. $\endgroup$ – Nick Gill Mar 7 '19 at 16:28
  • $\begingroup$ I meant that I thought it unlikely that there would be more than a small number of individual examples satisfying (1)-(4), although there could be a few more with $|\Lambda| = 3$ or $4$. I don't know the answer to your final question. If the stabilizer is to be large and to have a regular orbit, then the degree must be very large, and we soon reach the limits of brute force computation. One example I found that might be promising is $G = O_7(3)$ (the simple group) $H = S_4 \times S_6$, for which we get $|\Lambda| = 15$ with $S_6$ induced. $\endgroup$ – Derek Holt Mar 8 '19 at 13:32
  • $\begingroup$ Thanks Derek, that's a really interesting example. I will ponder... $\endgroup$ – Nick Gill Mar 8 '19 at 16:04

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