2023/02/27 by Marina Anagnostopoulou‐Merkouri, Peter J. Cameron, Anagnostopoulou-Merkouri, Marina +3
Computer Science · Mathematics · #20B05 #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2302.13703
openalex publication_date 2023/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A transitive permutation group G on a finite set Ω is said to be pre-primitive if every G-invariant partition of Ω 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.