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

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 $\mathrm{\Omega }$ and two distinct points $\alpha ,\beta \in \mathrm{\Omega }$ with $G_{\alpha \beta }⊴G_{\alpha }$ and $G_{\alpha \beta }\mathrm{\ne }1$, 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.