Two local conditions on the vertex stabiliser of arc-transitive graphs and their effect on the Sylow subgroups

TL;DR

This paper investigates specific local conditions on vertex stabilizers in arc-transitive graphs and demonstrates their implications on Sylow subgroups, establishing a new theorem for a broad class of such graphs.

Contribution

It introduces two local conditions on vertex stabilizers that lead to a Thompson-Wielandt-like theorem for many arc-transitive graphs, expanding understanding of their Sylow subgroups.

Findings

Subgroups fixing neighborhoods are p-groups for certain arcs.

Locally primitive, quasiprimitive, and semiprimitive graphs satisfy the conditions.

Counterexamples show the conditions do not always apply, with large composition factors.

Abstract

In this paper we study $G$-arc-transitive graphs $\Delta$ where the permutation group $G_x^{\Delta(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $\Delta(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $\Delta$, if $(x,y)$ is an arc of $\Delta$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $\Delta(x)$ and $\Delta(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-like theorem for a very large class of arc-transitive graphs. Furthermore, we give various families of $G$-arc-transitive graphs where our two local conditions do not apply and where $G_{x,y}^{[1]}$ has arbitrarily large composition factors.