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

TL;DR

This paper refines Sims' conjecture on the size of point stabilizers in finite primitive groups, providing new examples and progress, especially for groups of diagonal type, and addressing questions raised by Cameron and Fomin.

Contribution

It proposes a refined version of Sims' conjecture and constructs specific primitive groups of diagonal type to illustrate new properties and answer open questions.

Findings

Constructed primitive groups with specific stabilizer properties

Provided counterexamples related to Sims' conjecture

Addressed questions by Cameron and Fomin

Abstract

In 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 $\Omega$ and two distinct points $\alpha,\beta\in \Omega$ with $G_{\alpha\beta}\unlhd G_\alpha$ and $G_{\alpha\beta}\ne 1$, where $G_{\alpha}$ is the stabilizer of $\alpha$ in $G$ and $G_{\alpha\beta}$ is the stabilizer of $\alpha$ and $\beta$ in $G$. In particular, this example gives an answer to a question raised independently by Peter Cameron and by Alexander Fomin.