Pre-primitive permutation groups

Marina Anagnostopoulou-Merkouri, Peter J. Cameron*, Enoch Suleiman

*Corresponding author for this work

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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.
Original languageEnglish
Pages (from-to)695-715
JournalJournal of Algebra
Early online date17 Sept 2023
Publication statusPublished - 15 Dec 2023


  • Transitive permutation group
  • Invariant partition
  • Quasiprimitivity


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