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What are the possible sizes of a point stabilizer in a primitive permutation group, where the point stabilizer has an orbit of size 4?

In the tradition of subdegree 3 and subdegree 2, I wonder about subdegree 4. This would be the next exercise in the series 8B.5, 8B.6, 8B.7 on page 249 of Isaacs's Finite Group Theory textbook.

I have a rough proof that the point stabilizer has order of the form 2a3b, and (so far at least) all the examples I've looked at have a ≤ 5, b ≤ 2.

Is it true that the point stabilizer has order of the form 2a3b? If so, is it true that a ≤ 5, b ≤ 2?

In general, I think it is true that every prime divisor of the order of the point stabilizer is less than or equal to the size of the suborbit, but I don't have any real feel for how large the prime powers can be. My proof is silly-easy and doesn't use any graph theory.

I haven't yet had a chance to carefully read Sims (1967) [exact a for subdegree 3], Wong (1967) [exact structure for subdegree 3], Sims (1968) [exact a for subdegree 4, but assuming b=0?], Thompson (1970) [general control], or Cameron et al. (1983) [existence of general bound], so the answer could be fairly easy.

I think CPSS (1983) radically simplifies (maybe even trivializes) for subdegree 4, but Sims (1968) seems like only a partial answer. It's not clear to me if we know some simple (at least for subdegree 4) combinatorics problem equivalent to this problem.

  • Sims, Charles C. "Graphs and finite permutation groups." Math. Z. 95 (1967) 76–86. MR204509 DOI:10.1007/BF01117534
  • Wong, Warren J. "Determination of a class of primitive permutation groups." Math. Z. 99 (1967) 235–246. MR214653 DOI:10.1007/BF01112454
  • Sims, Charles C. "Graphs and finite permutation groups. II." Math. Z. 103 (1968) 276–281. MR225865 DOI:10.1007/BF01114994
  • Thompson, John G. "Bounds for orders of maximal subgroups." J. Algebra 14 (1970) 135–138. MR252500 DOI:10.1016/0021-8693(70)90117-1
  • Cameron, P. J., Praeger, C. E., Saxl, J., & Seitz, G. M. "On the Sims conjecture and distance transitive graphs." Bull. London Math. Soc., 15(5) (1983) 499–506. MR705530 DOI:10.1112/blms/15.5.499
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I just had a quick look at the papers. Sims [1967], Theorem 5.4 gets a bound for the case when $G_\alpha$ induces $A_4$ on the orbit, and Sims [1967] does it when the induced action is a 2-group. That appears to leave the case when the induces action is $S_4$ - on the other hand, Thompson [1970] states that Sims got a bound in all cases in these two papers, so I don't know! – Derek Holt Oct 24 '11 at 8:15

Ignoring CPSS, which I am not sure is helpful to find the exact upper bound, the next important references on this problem are (in chronological order) :

[1] W.L. Quirin, Primitive permutation groups with small orbitals, Math. Z. 122 (1971) 267–274.

[2] J. Wang, The primitive permutation groups with an orbital of length 4, Comm. Algebra 20 (1992) 889–921.

[3] C.H. Li, Z.P. Lu, D. Marusic, On primitive permutation groups with small suborbits and their orbital graphs, J. Algebra 279 (2004) 749-770.

The best is simply to read the introduction to [3]. It includes a short history and the full list of possible stabilisers is given, including a correction about the claimed existence of an example of order $2^43^6$.

It turns out that the order of a stabiliser divides either $2^5$ or $2^43^2$ and this is sharp. (So the answer to your question is "yes".)

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