On locally semiprimitive graphs and a theorem of Weiss

Abstract

In this paper we investigate graphs that admit a group acting arc-transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup. This type of semiprimitive group is a generalisation of an affine group. We show that if the graph has valency coprime to six, then there is a bound on the order of the vertex stabilisers depending on the valency alone. We also prove a detailed structure theorem for the vertex stabilisers in the remaining case. This is a contribution to an ongoing project to investigate the validity of the Poto\v{c}nik-Spiga-Verret Conjecture

An application of the Local C(G,T) Theorem to a conjecture of Weiss

“…As u and w are both neighbours of v, we have G [3] u ≤ G [1] vw . Therefore G [3] u ≤ Q w and hence Z w centralises G [3] u .…”

“…As u and w are both neighbours of v, we have G [3] u ≤ G [1] vw . Therefore G [3] u ≤ Q w and hence Z w centralises G [3] u . As G [3] u is G uinvariant, X centralises G [3] u and hence also G…”

“…is a primitive group containg an abelian regular subgroup and let |Γ(v)| = r ℓ for some prime r and positive integer ℓ. Then G Other recent and significant indications towards a positive solution to the Weiss Conjecture are given in [2,3,5,6,7,8].…”

Three local actions in 6‐valent arc‐transitive graphs

“…. Tutte's result [28] can then be rephrased as saying that transitive groups of degree 3 are graph-restrictive (with corresponding constant 16), while the famous Weiss conjecture [31] claims that every primitive permutation group is graph-restrictive; see [17] for a survey of results on graph-restrictiveness and [2,9,10,26] for more recent results.…”

Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

“…This class of permutation groups was introduced by Bereczky and Maróti [5] who were motivated by an interest in universal algebras and the structure of collapsing monoids. Semiprimitive groups have been the topic of several recent papers [11,12,35], motivated by the connection to the Weiss Conjecture in graph theory [30,36,39], via the Potočnik-Spiga-Verret Conjecture [24].…”