AllHamiltonianGroups - Maple Help
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GroupTheory

  

HamiltonianGroup

  

construct a finite Hamiltonian group

  

NumHamiltonianGroups

  

find the number of Hamiltonian groups of a given order

  

AllHamiltonianGroups

  

find all Hamiltonian groups of a given order

 

Calling Sequence

Parameters

Options

Description

Examples

Compatibility

Calling Sequence

HamiltonianGroup( n, k )

NumHamiltonianGroups( n )

AllHamiltonianGroups( n )

Parameters

n

-

a positive integer

k

-

a positive integer

Options

• 

formopt : option of the form form = "permgroup" or form = "fpgroup"

• 

outopt : option of the form output = "list" or output = "iterator"

Description

• 

A group is Hamiltonian if it is non-Abelian, and if every subgroup is normal. Every Hamiltonian group has the quaternion group as a direct factor, so the order of every finite Hamiltonian group is a multiple of .

• 

For a positive integer n, the NumHamiltonianGroups( n ) command returns the number of Hamiltonian groups of order n. (This is  if n is not a multiple of .)

• 

The HamiltonianGroup( n, k ) command returns the k-th Hamiltonian group of order n. An exception is raised if n is not a multiple of .

• 

The AllHamiltonianGroups( n ) command returns an expression sequence of all the Hamiltonian groups of order n, where n is a positive integer. Note that NULL is returned if n is not a multiple of .

• 

The HamiltonianGroup and AllHamiltonianGroups commands accept an option of the form form = F, where F may be either of the strings "permgroup" (the default), or "fpgroup".

• 

The AllHamiltonianGroups command accepts an option of the form output = "list" (the default) or output = "iterator". By default, a sequence of the Hamiltonian groups of order n is returned. If you pass the option output = "iterator" to AllHamiltonianGroups, then an iterator object is returned instead.

Examples

There is an unique Hamiltonian group of each -power greater than or equal to .

(1)

There are no Hamiltonian groups of order .

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Compatibility

• 

The GroupTheory[HamiltonianGroup], GroupTheory[NumHamiltonianGroups] and GroupTheory[AllHamiltonianGroups] commands were introduced in Maple 2019.

• 

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

See Also

GroupTheory

GroupTheory[IsHamiltonian]

GroupTheory[NumGroups]

 


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