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GroupTheory

  

IsHallPaigeGroup

  

determine whether a finite group has a complete mapping

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsHallPaigeGroup( G )

Parameters

G

-

a group

Description

• 

A permutation φ of a finite group G is said to be a complete mapping if the function psi defined by psig=`.`g,phig, for gG, 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

withGroupTheory:

IsHallPaigeGroupDihedralGroup5

false

(1)

IsHallPaigeGroupDihedralGroup10

true

(2)

IsHallPaigeGroupSymm4

true

(3)

IsHallPaigeGroupElementaryGroup3,3

true

(4)

IsHallPaigeGroupMathieuGroup22

true

(5)

IsHallPaigeGroupFreeGroup3

true

(6)

Compatibility

• 

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

• 

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

See Also

GroupTheory

GroupTheory[SylowSubgroup]