Graphs with automorphism groups admitting composition factors of bounded rank

Abstract

We prove a 1978 1978 conjecture of Richard Weiss in the case of groups with composition factors of bounded rank. Namely, we prove that there exists a function g : N × N → N g: \mathbb {N} \times \mathbb {N} \to \mathbb {N} such that, for Γ \Gamma a connected G G -vertex-transitive, G G -locally primitive graph of valency at most d d , if G G has no alternating groups of degree greater than r r as sections, then a vertex stabiliser in G G has size at most g ( r , d ) g(r,d) .