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

John Bray, Qi Cai, Peter Jephson Cameron, Pablo Spiga, Hua Zhang

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21 Citations (Scopus)
4 Downloads (Pure)


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.
Original languageEnglish
Pages (from-to)27-42
JournalJournal of Algebra
Early online date7 Mar 2019
Publication statusPublished - Mar 2020


  • Automata
  • Complete mappings
  • Graphs
  • Hall-Paige conjecture
  • Orbitals
  • Primitive groups
  • Separating groups
  • Synchronizing groups
  • Transformation groups


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