GroupTheory/IsHallPaigeGroup - Maple Help

GroupTheory

 IsHallPaigeGroup
 determine whether a finite group has a complete mapping

 Calling Sequence IsHallPaigeGroup( G )

Parameters

 G - a group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{IsHallPaigeGroup}\left(\mathrm{DihedralGroup}\left(5\right)\right)$
 ${\mathrm{false}}$ (1)
 > $\mathrm{IsHallPaigeGroup}\left(\mathrm{DihedralGroup}\left(10\right)\right)$
 ${\mathrm{true}}$ (2)
 > $\mathrm{IsHallPaigeGroup}\left(\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsHallPaigeGroup}\left(\mathrm{ElementaryGroup}\left(3,3\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{IsHallPaigeGroup}\left(\mathrm{MathieuGroup}\left(22\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IsHallPaigeGroup}\left(\mathrm{FreeGroup}\left(3\right)\right)$
 ${\mathrm{true}}$ (6)

Compatibility

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