A complete map for a group G is a permutation phi: G --> G such that g bar right arrow g phi (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\ less than or equal to 2, G' congruent to SL(2, q) for some odd prime power q > 5 and if G is not a perfect group then G/Z(G') congruent to PGL(2, q).
Minimal counterexamples to a conjecture of Hall and Paige
GAVIOLI, NORBERTO
2001-01-01
Abstract
A complete map for a group G is a permutation phi: G --> G such that g bar right arrow g phi (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\ less than or equal to 2, G' congruent to SL(2, q) for some odd prime power q > 5 and if G is not a perfect group then G/Z(G') congruent to PGL(2, q).File in questo prodotto:
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