Graphs with automorphism groups admitting composition factors of bounded rank (Q2845047)

A graph \(\Gamma\) is called \(G\)-vertex-transitive if \(G\) is a group of automorphisms of \(\Gamma\) with a single orbit on the vertex set of \(\Gamma\), and then \(G\)-locally primitive if the permutation group induced on the neighbourhood \(\Gamma(v)\) of a vertex \(v\) by the stabiliser \(G_v\) of \(v\) is primitive. In 1978 it was conjectured by Richard Weiss that in the \(G\)-locally primitive case, the order of \(G_v\) is bounded above by a function of the valency of \(\Gamma\); see [\textit{R. Weiss}, Colloq. Math. Soc. Janos Bolyai 25, 827--847 (1981; Zbl 0475.05040)].NEWLINENEWLINENEWLINE In this paper, the authors prove a weaker form of the Weiss conjecture, namely one that applies for groups \(G\) with composition factors of bounded rank.NEWLINENEWLINESpecifically, they consider the class BCP\((r)\) of finite groups \(G\) for which no section \(H/K\) of \(G\) is isomorphic to Alt\((r \!+\!1)\), as defined by \textit{L. Babai} et al. [J. Algebra 79, 161--168 (1982; Zbl 0493.20001)], and prove that there exists a 2-variable function \(g\) defined on the positive integers such that if \(G\) is a BCP\((r)\)-group and \(\Gamma\) is a connected \(G\)-vertex-transitive, \(G\)-locally primitive graph of valency at most \(d\), then \(|G_v| \leq g(r, d)\) for every vertex \(v\).NEWLINENEWLINENEWLINEIn fact they prove a slightly more general theorem, using recent work of Praeger, Spiga and Verret to reduce to the case of finite simple groups, and recent work on growth in simple groups (by Pyber and Szabó and by Breuillard, Green and Tao).