Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture. I

This is the first in a series of papers whose results imply the validity of a strong version of the Sims conjecture on finite primitive permutation groups from the authors' article “Stabilizers of graph's vertices and a strengthened version of the Sims conjecture”, Dokl. Math. 59(1), 113–115 (1999). In this paper, the case of non almost simple primitive groups and the case of primitive groups with alternating socle are considered.