Minimal counterexamples to a conjecture of Hall and Paige

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 \( G/Z(G')\cong \rm{PGL}(2,\it{q}) \).