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

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups L<inf>1</inf> and L<inf>2</inf> with L<inf>1</inf> not semiprimitive, we construct an infinite family of rank two amalgams of permutation type [L<inf>1</inf>, L<inf>2</inf>] 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<inf>1</inf>, L<inf>2</inf>] 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 [L<inf>1</inf>, . . ., L<inf>k</inf>] only in the trivial case where each of L<inf>1</inf>, . . ., L<inf>k</inf> is regular.