  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.
