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