Pre-primitive permutation groups
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A transitive permutation group G on a finite set Omega is said to be pre-primitive if every G-invariant partition of Omega is the orbit partition of a subgroup of G. It follows that pre-primitivity and quasiprimitivity are logically independent (there are groups satisfying one but not the other) and their conjunction is equivalent to primitivity. Indeed, part of the motivation for studying pre-primitivity is to investigate the gap between primitivity and quasiprimitivity. We investigate the pre-primitivity of various classes of transitive groups including groups with regular normal subgroups, direct and wreath products, and diagonal groups. In the course of this investigation, we describe all G-invariant partitions for various classes of permutation groups G. We also look briefly at conditions similarly related to other pairs of conditions, including transitivity and quasiprimitivity, k-homogeneity and k-transitivity, and primitivity and synchronization.
Anagnostopoulou-Merkouri , M , Cameron , P J & Suleiman , E 2023 , ' Pre-primitive permutation groups ' , Journal of Algebra , vol. 636 , pp. 695-715 . https://doi.org/10.1016/j.jalgebra.2023.09.012
Journal of Algebra
Copyright © 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
DescriptionFunding: The first author was funded by a StARIS research internship from the University of St Andrews.
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