Pre-primitive permutation groups
| Autor: | Anagnostopoulou-Merkouri, Marina, Cameron, Peter J., Suleiman, Enoch |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: To appear in Journal of Algebra |
| Databáze: | arXiv |
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