GroupTheory - Maple Programming Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Group Theory : GroupTheory/IsHamiltonian

GroupTheory

  

IsHamiltonian

  

attempt to determine whether a group is Hamiltonian

  

IsDedekind

  

attempt to determine whether a group is Dedekind

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

IsDedekind( G )

IsHamiltonian( G )

Parameters

G

-

a permutation group

Description

• 

A group G is Dedekind if every subgroup of G is normal in G. Every Abelian group is obviously a Dedekind group, but non-Abelian Dedekind groups exist.

• 

A group G is Hamiltonian if it is a non-commutative Dedekind group.

• 

The IsDedekind( G ) command attempts to determine whether the group G is Dedekind.  It returns true if G is Dedekind and returns false otherwise.

• 

The IsHamiltonian( G ) command attempts to determine whether the group G is Hamiltonian, returning true if G is Hamiltonian, and false otherwise.

• 

The smallest Hamiltonian group is the quaternion group of order 8.

Examples

withGroupTheory:

IsHamiltonianQuaternionGroup

true

(1)

andmapIsNormal,convertSubgroupLatticeQuaternionGroup,list,QuaternionGroup

true

(2)

The fact that this group is Hamiltonian is visible from the subgroup lattice:

DrawSubgroupLatticeQuaternionGroup

IsDedekindQuaternionGroup

true

(3)

IsHamiltonianCyclicGroup10

false

(4)

IsDedekindCyclicGroup10

true

(5)

IsDedekindDihedralGroup4

false

(6)

IsHamiltonianDihedralGroup4

false

(7)

andmapIsNormal,convertSubgroupLatticeDihedralGroup4,list,DihedralGroup4

false

(8)

You can see that the dihedral group of order 8 is not Hamiltonian by looking at its subgroup lattice.

DrawSubgroupLatticeDihedralGroup4

IsHamiltonianSmallGroup256,56084

true

(9)

IsDedekindSmallGroup256,56085

false

(10)

Compatibility

• 

The GroupTheory[IsHamiltonian] and GroupTheory[IsDedekind] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[DihedralGroup]

GroupTheory[IsAbelian]

GroupTheory[IsNormal]

GroupTheory[QuaternionGroup]

GroupTheory[SmallGroup]