A Construction of Imprimitive Groups of Rank 4 or 5

Abstract

Let G be a transitive permutation group acting on a finite set Ω. For a point α of Ω, the set of the images of G acting on α is called the orbit of α under G and is denoted by α^G, and the set of elements in G which fix α is called the stabilizer of α in G and is denoted by G_α. We can get some new orbits by using the natural action of the stabilizer G_α on Ω, and then we can define the suborbit of G. The suborbits of G on Ω are defined as the orbits of a point stabilizer on Ω. The number of suborbits is called the rank of G and the length of suborbits is called the subdegree of G. For finite primitive groups, the study of the rank and subdegrees of group has a long history. In this paper, we construct a class of imprimitive permutation groups of rank 4 or 5 by using imprimitive action and product action of wreath product, determine the number and the length of the suborbits, and extend the results to imprimitive permutation groups of rank m+1 and 2ⁿ+1, where m and n are positive integers.