A permutation $\mathrm{\phi }$ of a finite group $G$ is said to be a complete mapping if the function $\mathrm{psi}$ defined by $\mathrm{psi}\left(g\right)\=\mathrm{`.`}\left(g\,\mathrm{phi}\left(g\right)\right)$, for $g\in G$, 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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{IsHallPaigeGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$

$\mathrm{false}$

$\mathrm{IsHallPaigeGroup}\left(\mathrm{DihedralGroup}\left(10\right)\right)$

$\mathrm{true}$

$\mathrm{IsHallPaigeGroup}\left(\mathrm{Symm}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{IsHallPaigeGroup}\left(\mathrm{ElementaryGroup}\left(3\,3\right)\right)$

$\mathrm{true}$

$\mathrm{IsHallPaigeGroup}\left(\mathrm{MathieuGroup}\left(22\right)\right)$

$\mathrm{true}$

$\mathrm{IsHallPaigeGroup}\left(\mathrm{FreeGroup}\left(3\right)\right)$

$\mathrm{true}$

The GroupTheory[IsHallPaigeGroup] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.