Imprimitive Permutations in Primitive Groups
J. Ara\'ujo, J. P. Ara\'ujo, P. J. Cameron, T. Dobson, A. Hulpke, P., Lopes

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,…
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Taxonomy
TopicsFinite Group Theory Research · Coding theory and cryptography · graph theory and CDMA systems
