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Archiv der Mathematik

, Volume 77, Issue 3, pp 209–214 | Cite as

Minimal counterexamples to a conjecture of Hall and Paige <!-

  • F. Dalla Volta
  • N. Gavioli
Article

Abstract.

A complete map for a group G is a permutation \( \varphi\colon G\to G \) such that \( g\mapsto g\varphi(g) \) is still a permutation of G. A conjecture of M. Hall and L. J. Paige states that every finite group whose Sylow 2-subgroup is non-trivial and non-cyclic admits a complete map. In the present paper it is proved that a potential counterexample G of minimal order to this conjecture either is almost simple or G has only one involution, the Sylow 2-subgroups of G are quaternionic, \( |G/G'|\leqq 2$, $G'\cong \ SL(2,q) \) for some odd prime power \( q<5 \) and if G is not a perfect group then \( G/Z(G')\cong \rm{PGL}(2,\it{q}) \).

Keywords

Finite Group Prime Power Minimal Order Minimal Counterexample Perfect Group 

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • F. Dalla Volta
    • 1
  • N. Gavioli
    • 2
  1. 1.Dipartimento di Matematica e Applicazioni, Edificio U7, Università degli Studi di Milano–Bicocca, Via Bicocca degli Arcimboldi 8, I-20126 Milano – Italy, dallavolta@matapp.unimib.itItaly
  2. 2.Dipartimento di matematica pura ed applicata, Università degli Studi dell'Aquila, Via Vetoio, I-67010 Coppito (L'Aquila) – Italy, gavioli@univaq.itItaly

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