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Imprimitive permutations in primitive groups

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Joao Araújo, Joao Pedro Araújo, Peter Jephson Cameron, Ted Dobson, Alexander Hulpke, Pedro Lopes

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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 we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems.


Original languageEnglish
Pages (from-to)396-416
JournalJournal of Algebra
Early online date2 May 2017
StatePublished - 15 Sep 2017

    Research areas

  • Primitive groups, Imprimitive groups, GAP, Permutation type

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