×

The Hall-Paige conjecture, and synchronization for affine and diagonal groups. (English) Zbl 1485.20002

Summary: 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 \(\mathrm{J}_4)\).
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.

MSC:

20B15 Primitive groups
05E30 Association schemes, strongly regular graphs
20D08 Simple groups: sporadic groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20M20 Semigroups of transformations, relations, partitions, etc.
20M35 Semigroups in automata theory, linguistics, etc.
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aljohani, Mohammed; Bamberg, John; Cameron, Peter J., Synchronization and separation in the Johnson schemes, Port. Math., 74, 3, 213-232 (2017) · Zbl 1423.05196
[2] Araújo, João; Cameron, Peter J.; Steinberg, Benjamin, Between primitive and 2-transitive: synchronization and its friends, Eur. Math. Soc. Surv., 4, 101-184 (2017) · Zbl 1402.68124
[3] Rosemary A. Bailey, Peter J. Cameron, Cheryl E. Praeger, Csaba Schneider, Diagonal structures, in preparation. · Zbl 1509.20003
[4] Ball, Simeon; Govaerts, Patrick; Storme, Leo, On ovoids of parabolic quadrics, Des. Codes Cryptogr., 38, 131-145 (2006) · Zbl 1172.51300
[5] Bosma, Wieb; Cannon, John; Playoust, Catherine, The Magma algebra system. I. The user language, J. Symbolic Comput., 24, 235-265 (1997) · Zbl 0898.68039
[6] Bray, J. N., An improved method for generating the centralizer of an involution, Arch. Math., 74, 241-245 (2000) · Zbl 0956.20022
[7] Qi Cai, Hua Zhang, Synchronizing groups of affine type are separating, preprint.
[8] Dixon, John D.; Mortimer, Brian, Permutation Groups (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0951.20001
[9] Evans, Anthony B., The admissibility of sporadic simple groups, J. Algebra, 321, 1, 105-116 (2009) · Zbl 1166.20012
[10] GAP - Groups, Algorithms, and Programming (2018), Version 4.10.0
[11] Jr., Marshall Hall; Paige, L. J., Complete mappings of finite groups, Pacific J. Math., 5, 541-549 (1955) · Zbl 0066.27703
[12] Kleidman, Peter B.; Wilson, Robert A., The maximal subgroups of \(J_4\), Proc. Lond. Math. Soc., 56, 484-510 (1988) · Zbl 0619.20004
[13] Praeger, Cheryl E.; Schneider, Csaba, Permutation Groups and Cartesian Decompositions, London Math. Soc. Lecture Note Series, vol. 449 (2018), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1428.20002
[14] Wilcox, Stewart, Reduction of the Hall-Paige conjecture to sporadic simple groups, J. Algebra, 321, 5, 1407-1428 (2009) · Zbl 1172.20024
[15] Wilson, Robert; Walsh, Peter; Tripp, Jonathan; Suleiman, Ibrahim; Parker, Richard; Norton, Simon; Nickerson, Simon; Linton, Steve; Bray, John; Abbott, Rachel, Atlas of Finite Group Representations - Version 3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.