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

Original · EN

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups L₁ and L₂ with L₁ not semiprimitive, we construct an infinite family of rank two amalgams of permutation type [L₁,L₂] 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 [L₁,L₂] graphs. We also consider the corresponding question for amalgams of rank k≥ 3. We completely resolve this by showing that the order of the Borel subgroup is bounded by the permutation type [L₁,...,Lₖ] only in the trivial case where each of L₁,...,Lₖ is regular.