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Erschienen in:
2018 | OriginalPaper | Buchkapitel
6. Minimal Counterexamples to the Hall-Paige Conjecture
verfasst von : Anthony B. Evans
Erschienen in: Orthogonal Latin Squares Based on Groups
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Abstract
If the Hall-Paige conjecture is false, what might a minimal counterexample look like? This question was first raised by Aschbacher in 1990. He proved that any minimal counterexample must be “close” to being a simple group. In 1992 Evans tried to improve on this by extending complete mappings of H, a subgroup of G of index 2, to complete mappings of G; and complete mappings of G∕H, \(H\cong \mathbb {Z}_2\), to complete mappings of G. Evans’ proofs require that certain technical conditions hold for the extensions to work. In 2001 Dalla Volta and Gavioli improved on Aschbacher’s reduction and in 2009, using elementary techniques, Wilcox showed that any minimal counterexample to the Hall-Paige conjecture must be a nonabelian finite simple group. We will present these reductions in this chapter.