On the Burness-Giudici conjecture

Abstract

Let G be a permutation group on a set . A subset of is a base for G if its pointwise stabilizer in G is trivial. By b(G) we denote the size of the smallest base of G. Every permutation group with b(G) = 2 contains some regular suborbits. It is conjectured by Burness-Giudici that every primitive permutation group G with b(G) = 2 has the property that if then , where Γ is the union of all regular suborbits of G relative to α. An affirmative answer of the conjecture has been shown for many sporadic simple groups and some alternating groups, but it is still open for simple groups of Lie-type. The first candidate of an infinite family of simple groups of Lie-type we should work on might be , where . In this manuscript, we show the correctness of the conjecture for all the primitive groups with socle , see Theorem 1.3.