A class of semiprimitive groups that are graph‐restrictive

Abstract

Abstract. We prove that an infinite family of semiprimitive groups are graph-restrictive. This adds to the evidence for the validity of the PSV Conjecture and increases the minimal imprimitive degree for which this conjecture is open to 12. Our result can be seen as a generalisation of the well-known theorem of Tutte on cubic graphs. The proof uses the amalgam method, adapted to this new situation.

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

“…We conclude that G [2] uv = G [2] u ∩G [2] v is centralised by H v and H u . The Hauptlemma yields G [2] uv = 1, a contradiction.…”

“…Assume the notation in Lemma 2.7. Since G [1] uv is a p-group, we see that G [2] v is a p-group, and hence G [2] v ≤ Q v . If G [2] uv = 1, then the lemma follows immediately and hence we assume that G [2] uv = 1.…”

“…By arc-transitivity we have Z v Q u , and hence a symmetric argument gives that H u := Z x v | x ∈ G u centralises G [2] u and acts transitively on Γ(u). We conclude that G [2] uv = G [2] u ∩G [2] v is centralised by H v and H u . The Hauptlemma yields G [2] uv = 1, a contradiction.…”

“…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].…”

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].…”