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

© 2015 Elsevier Inc. 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.