On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs
Morgan, Luke ; Spiga, Pablo ; Verret, Gabriel
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 ▫$[L_1, \ldots, L_k]$▫ only in the trivial case where each of ▫$L_1, \ldots, L_k$▫ is regular.
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