Arc-transitive digraphs with quasiprimitive local actions

2016/10/19 by Giudici, Michael, Glasby, S. P., Li, Cai Heng +1 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1610.06222

Abstract

Let Γ be a finite G-vertex-transitive digraph. The in-local action of (Γ,G) is the permutation group L_- induced by the vertex-stabiliser on the set of in-neighbours of v. The out-local action L₊ is defined analogously. Note that L_- and L₊ may not be isomorphic. We thus consider the problem of determining which pairs (L_-,L₊) are possible. We prove some general results, but pay special attention to the case when L_- and L₊ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.

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