Imprimitive Permutations in Primitive Groups

TL;DR

This paper investigates the structure of primitive permutation groups that are unions of imprimitive groups, establishing infinite families, proposing a new hierarchy, and providing algorithms and open problems for further research.

Contribution

It introduces a new hierarchy for primitive groups based on their relation to imprimitive groups and provides algorithms and computational tools for analysis.

Findings

Existence of infinite families of primitive groups within unions of imprimitive groups

Development of algorithms and GAP implementation for permutation analysis

Resolution of some open problems in the theory of primitive groups

Abstract

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition to the previous results and hierarchy, we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems.