Classifying primitive solvable permutation groups of rank 5 and 6

Abstract

Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let G 0 G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of G 0 G_{0} in Ω, including the trivial orbit { α } \{\alpha\} . In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.