On automorphism groups of bi-quasiprimitive 2-arc-transitive graphs.

Let Γ be a connected (X , 2) -arc-transitive bipartite graph with bi-parts Δ 0 and Δ 1 , where X ≤ Aut (Γ). Let X + be the subgroup of X fixing Δ 0 setwise. In this paper, we first prove that if X + is primitive and faithful on Δ 0 and Δ 1 , then the actions of X + on Δ 0 and Δ 1 are primitive of the same type X with X ∈ { HA , SA , PA }. This is then used to prove that if X + is quasiprimitive on Δ 0 of type HA or TW, then either soc (X +) ⊴ Aut (Γ) , or Γ ≅ K q , q (q a prime power) or Γ ≅ K 2 r , 2 r − 2 r K 2 (r ≥ 2). This confirms a conjecture of C. H. Li from 2008. [ABSTRACT FROM AUTHOR]