Imprimitive permutations in primitive groups
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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.
Araújo , J , Araújo , J P , Cameron , P J , Dobson , T , Hulpke , A & Lopes , P 2017 , ' Imprimitive permutations in primitive groups ' Journal of Algebra , vol 486 , pp. 396-416 . DOI: 10.1016/j.jalgebra.2017.03.043
Journal of Algebra
© 2017, Elsevier, Inc. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.sciencedirect.com / https://doi.org/10.1016/j.jalgebra.2017.03.043
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