determine whether a finite group has a complete mapping
IsHallPaigeGroup( G )
A permutation φ of a finite group G is said to be a complete mapping if the function psi defined by psi⁡g=`.`⁡g,phi⁡g, for g∈G, is also bijective.
In 1955, M. Hall and L. J. Paige conjetured that a finite group has a complete mapping if, and only if, its Sylow 2-subgroups are non-cyclic, and proved the equivalence for soluble groups, as well as for the symmetric and alternating groups. (Paige had earlier observed already that groups of odd order have a complete mapping, as the identity mapping on the group will serve.) The conjecture was finally settled completely in published form in November of 2018.
A group G is called a Hall-Paige group if it has a complete mapping in the sense of Hall and Paige.
The IsHallPaigeGroup( G ) command attempts to determine whether the group G is a Hall-Paige group. It returns true if G is a Hall-Paige group and returns false otherwise.
The GroupTheory[IsHallPaigeGroup] command was introduced in Maple 2019.
For more information on Maple 2019 changes, see Updates in Maple 2019.
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