A generalization of Sims’ conjecture for finite primitive groups and two point stabilizers in primitive groups

Pablo Spiga

doi:10.1515/jgth-2021-0032

Abstract

AbstractIn this paper, we propose a refinement of Sims’ conjecture concerning the cardinality of the point stabilizers in finite primitive groups, and we make some progress towards this refinement. In this process, when dealing with primitive groups of diagonal type, we construct a finite primitive group 𝐺 on Ω and two distinct pointsα,β∈Ω\alpha,\beta\in\OmegawithGα⁢β⊴GαG_{\alpha\beta}\unlhd G_{\alpha}andGα⁢β≠1G_{\alpha\beta}\neq 1, whereGαG_{\alpha}is the stabilizer of 𝛼 in 𝐺 andGα⁢βG_{\alpha\beta}is the stabilizer of 𝛼 and 𝛽 in 𝐺. In particular, this example gives an answer to a question raised independently by Cameron and by Fomin in the Kourovka Notebook.

Highlights

Let G be a finite primitive group acting on a set Ω and let α ∈ Ω
In this paper we propose a strengthening of Sims conjecture that in our opinion captures the structure of point stabilizers of primitive groups to a finer degree
Let G be a finite primitive group acting on a set Ω and let α and β be two elements of Ω

Summary

Introduction

Let G be a finite primitive group acting on a set Ω and let α ∈ Ω. There exists a function g : N → N such that, if G is a finite primitive group with a suborbit βGα of length d > 1, either (i): Gαβ has order at most g(|Gαβ : G+α [1]|), or (ii): G is in a well-described and well-determined list of exceptions. It is quite unfortunate, that we cannot omit alternative (ii) in Conjecture 1.2.

Basic results

Primitive groups of SD type

The example for the Cameron and Fomin question