We show that the sporadic simple group M(22), the exceptional group of Lie type 2 E 6 (2) and their automorphism groups are uniquely determined by the approximate structure of the centralizer of an element of order 3 together with some information about the fusion of this element in the group.
Let G be a finite group and Γ a G-symmetric graph. Suppose that G is imprimitive on V(Γ) with B a block of imprimitivity and B := {B g ; g ∈ G} a system of imprimitivity of G on V(Γ). Define Γ B to be the graph with vertex set B such that two blocks B, C ∈ B are adjacent if and only if there exists at least one edge of Γ joining a vertex in B and a vertex in C. Xu and Zhou ['Symmetric graphs with 2-arc-transitive quotients', J. Aust. Math. Soc. 96 (2014), 275-288] obtained necessary conditions under which the graph Γ B is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to Γ and B and show that there is a unique graph corresponding to this case.2010 Mathematics subject classification: primary 05C25; secondary 05E18.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.