The Hall–Paige conjecture, and synchronization for affine and diagonal groups
MetadataShow full item record
Altmetrics Handle Statistics
Altmetrics DOI Statistics
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 J , Spiga , P & Zhang , H 2020 , ' The Hall–Paige conjecture, and synchronization for affine and diagonal groups ' , Journal of Algebra , vol. 545 , pp. 27-42 . https://doi.org/10.1016/j.jalgebra.2019.02.025
Journal of Algebra
© 2019, Elsevier, Inc. This work has been made available online in accordance with the publisher's policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1016/j.jalgebra.2019.02.025
Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.
Showing items related by title, author, creator and subject.
Coutts, Hannah Jane (University of St Andrews, 2011-11) - ThesisPart I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each ...
Plötner, Maria; Over, Harriet; Carpenter, Malinda; Tomasello, Michael (2016-03-24) - Journal articleTo date, developmental research on groups has focused mainly on in-group biases and intergroup relations. However, little is known about children’s general understanding of social groups and their perceptions of different ...
Wegner, Alexander (University of St Andrews, 1992) - ThesisComputational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting ...