On Cameron’s question about primitive permutation groups with stabilizer of two points that is normal in the stabilizer of one of them

Abstract

Assume that G is a primitive permutation group on a finite set X, x ∈ X, y ∈ X \ {x}, and Gx,y\(\underline \triangleleft \)Gx. P. Cameron raised the question about the validity of the equality Gx,y = 1 in this case. The author proved earlier that, if soc(G) is not a direct power of an exceptional group of Lie type, then Gx,y = 1. In the present paper, we prove that, if soc(G) is a direct power of an exceptional group of Lie type distinct from E6(q), 2E6(q), E7(q), and E8(q), then Gx,y = 1.