On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

TL;DR

This paper investigates the structure of Borel subgroups in group amalgams related to permutation groups, demonstrating unbounded Borel subgroup orders in certain cases and establishing bounds in trivial cases, with applications to locally-transitive graphs.

Contribution

It constructs infinite families of amalgams with increasing Borel subgroup orders and characterizes when Borel subgroup orders are bounded based on permutation group properties.

Findings

Unbounded Borel subgroup orders for non-semiprimitive groups

Bounded Borel subgroup orders only when all permutation groups are regular

No upper limit on edge-stabilizer orders in certain locally-transitive graphs

Abstract

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite family of rank two amalgams of permutation type $[L_1,L_2]$ and Borel subgroups of strictly increasing order. As an application, we show that there is no bound on the order of edge-stabilisers in locally $[L_1,L_2]$ graphs. We also consider the corresponding question for amalgams of rank $k\geq 3$. We completely resolve this by showing that the order of the Borel subgroup is bounded by the permutation type $[L_1,...,L_k]$ only in the trivial case where each of $L_1,...,L_k$ is regular.