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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"

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 8.

• 

For a positive integer n, the NumHamiltonianGroups( n ) command returns the number of Hamiltonian groups of order n. (This is 0 if n is not a multiple of 8.)

• 

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 8.

• 

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 8.

• 

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".

Examples

withGroupTheory:

There is an unique Hamiltonian group of each 2-power greater than or equal to 8.

seqNumHamiltonianGroups2i,i=1..20

0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

(1)

There are no Hamiltonian groups of order 25.

NumHamiltonianGroups25

0

(2)

NumHamiltonianGroups432

3

(3)

GHamiltonianGroup432,2

G14,15,16,17,18,19,20,21,22,11,12,13,9,10,1,2,3,45,6,8,7,1,5,3,82,7,4,6

(4)

IsHamiltonianG

true

(5)

AllHamiltonianGroups432,form=fpgroup

i,j,_g,_a1,_a2,_a3_g2,_a13,_a23,_a33,i4,i2j2,iji-1j,_a1-1_g-1_a1_g,_a1-1i-1_a1i,_a1-1j-1_a1j,_a2-1_a1-1_a2_a1,_a2-1_g-1_a2_g,_a2-1i-1_a2i,_a2-1j-1_a2j,_a3-1_a1-1_a3_a1,_a3-1_a2-1_a3_a2,_a3-1_g-1_a3_g,_a3-1i-1_a3i,_a3-1j-1_a3j,i-1_g-1i_g,j-1_g-1j_g,i,j,_g0,_a1,_a2_g02,_a13,i4,i2j2,iji-1j,_a1-1_g0-1_a1_g0,_a1-1i-1_a1i,_a1-1j-1_a1j,_a2-1_a1-1_a2_a1,_a2-1_g0-1_a2_g0,_a2-1i-1_a2i,_a2-1j-1_a2j,i-1_g0-1i_g0,j-1_g0-1j_g0,_a29,i,j,_g1,_a1_g12,i4,i2j2,iji-1j,_a1-1_g1-1_a1_g1,_a1-1i-1_a1i,_a1-1j-1_a1j,i-1_g1-1i_g1,j-1_g1-1j_g1,_a127

(6)

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]