On the order of Borel subgroups of group amalgams and an application to 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.