The Hall–Paige conjecture asserts that a finite group has a complete mapping if and only if its Sylow subgroups are not cyclic. The conjecture is now proved, and one aim of this paper is to document the final step in the proof (for the sporadic simple group J4). We apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non-synchronizing. We also give the simpler proof that, for groups of affine type, or simple diagonal type with two socle factors, synchronization and separation are equivalent. Synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than 2-homogeneity, the second of these being stronger than the first. Empirically it has been found that groups which are synchronizing but not separating are rather rare. It follows from our results that such groups must be primitive of almost simple type.

Bray, J., Cai, Q., Cameron, P., Spiga, P., Zhang, H. (2020). The Hall–Paige conjecture, and synchronization for affine and diagonal groups. JOURNAL OF ALGEBRA, 545(1 March 2020), 27-42 [10.1016/j.jalgebra.2019.02.025].

The Hall–Paige conjecture, and synchronization for affine and diagonal groups

Spiga P.;
2020

Abstract

The Hall–Paige conjecture asserts that a finite group has a complete mapping if and only if its Sylow subgroups are not cyclic. The conjecture is now proved, and one aim of this paper is to document the final step in the proof (for the sporadic simple group J4). We apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non-synchronizing. We also give the simpler proof that, for groups of affine type, or simple diagonal type with two socle factors, synchronization and separation are equivalent. Synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than 2-homogeneity, the second of these being stronger than the first. Empirically it has been found that groups which are synchronizing but not separating are rather rare. It follows from our results that such groups must be primitive of almost simple type.
Articolo in rivista - Articolo scientifico
Automata; Complete mappings; Graphs; Hall–Paige conjecture; Orbitals; Primitive groups; Separating groups; Synchronizing groups; Transformation semigroups;
English
27
42
16
Bray, J., Cai, Q., Cameron, P., Spiga, P., Zhang, H. (2020). The Hall–Paige conjecture, and synchronization for affine and diagonal groups. JOURNAL OF ALGEBRA, 545(1 March 2020), 27-42 [10.1016/j.jalgebra.2019.02.025].
File in questo prodotto:
File Dimensione Formato  
1-s2.0-S0021869319301140-main.pdf

Solo gestori archivio

Tipologia di allegato: Publisher’s Version (Version of Record, VoR)
Dimensione 363.27 kB
Formato Adobe PDF
363.27 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/345857
Citazioni
  • Scopus 16
  • ???jsp.display-item.citation.isi??? 16
Social impact