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

JOURNAL OF ALGEBRA

卷:545

Article

Bray, John N.¹  Cai, Qi²  Cameron, Peter J.³  Spiga, Pablo⁴  Zhang, Hua² 

[1] Queen Mary Univ London, Sch Math Sci, Mile End Rd, London E1 4NS, England

[2] Yunnan Normal Univ, 768 Juxian St,Chenggong Campus, Kunming 651010, Yunnan, Peoples R China

[3] Univ St Andrews, Sch Math & Stat, St Andrews KY16 9SS, Fife, Scotland

[4] Univ Milano Bicocca, Dipartimento Matemat Applicaz, Via Cozzi 55, I-20125 Milan, Italy

关键词: Automata;    Complete mappings;    Graphs;    Hall-Paige conjecture;    Orbitals;    Primitive groups;    Separating groups;    Synchronizing groups;    Transformation semigroups;   

DOI  :  10.1016/j.jalgebra.2019.02.025

来源: Elsevier

【 摘 要 】

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 J(4)). 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. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free