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GroupTheory

  

QuaternionGroup

  

construct the quaternion group of order 8

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

QuaternionGroup()

QuaternionGroup( f )

Parameters

f

-

(optional) equation of the form form = "permgroup" (default) or form = "fpgroup"

Description

• 

The quaternion group is one of the two non-abelian groups of order 8, (the other being the dihedral group of degree 4). It is notable because it is an example of a Hamiltonian group - every one of its subgroups is normal - and it appears as a subgroup of every non-Abelian Hamiltonian group.

• 

The QuaternionGroup() command returns a quaternion group, either as a permutation group (the default), or as a finitely presented group.

• 

You can pass the option 'form' = "fpgroup" or 'form' = "permgroup" to cause the QuaternionGroup command to return a group of the indicated class.

• 

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

withGroupTheory:

QuaternionGroup

GroupTheory:-QuaternionGroupform=permgroup

(1)

QuaternionGroupform=permgroup

GroupTheory:-QuaternionGroupform=permgroup

(2)

QuaternionGroupform=fpgroup

GroupTheory:-QuaternionGroupform=fpgroup

(3)

GroupOrderQuaternionGroup

8

(4)

There are only two non-Abelian groups of order eight.

SearchSmallGroupsorder=8,abelian=false

8,3,8,4

(5)

One of these is the Quaternion group.

IdentifySmallGroupQuaternionGroup

8,4

(6)

The dihedral group of order 8 (and degree 4) is the other group of order 8. It is not isomorphic to the quaternion group.

AreIsomorphicQuaternionGroup,DihedralGroup4

false

(7)

However, the quaternion and dihedral groups of order eight do have the same character tables.

ctQCharacterTableQuaternionGroup

GroupTheory:-CharacterTablemodule...end module

(8)

ctDCharacterTableDihedralGroup4

GroupTheory:-CharacterTablemodule...end module

(9)

EqualEntriesGetMatrixctQ,GetMatrixctD

true

(10)

(Notice, however, that the quaternion group has a single conjugacy class of involutions, while the dihedral group of order 8 has three conjugacy classes of involutions.)

GraphTheory:-DrawGraphCayleyGraphQuaternionGroup

DrawCayleyTableQuaternionGroup

DrawSubgroupLatticeQuaternionGroup

The quaternion group is an example of a Hamiltonian group - every one of its subgroups is normal. This is evident from the subgroup lattice diagram above; alternatively, Hamiltonicity can be demonstrated, as follows.

andmapIsNormal,convertSubgroupLatticeQuaternionGroup,list,QuaternionGroup

true

(11)

Like the dihedral group of order 8, the quaternion group is an extra-special 2-group.

mapGroupOrder,Centre,DerivedSubgroup,FrattiniSubgroupQuaternionGroup

2,2,2

(12)

map`@`op,Generators,Centre,DerivedSubgroup,FrattiniSubgroupQuaternionGroup

module...end module

(13)

Compatibility

• 

The GroupTheory[QuaternionGroup] command was introduced in Maple 17.

• 

For more information on Maple 17 changes, see Updates in Maple 17.

See Also

GroupTheory[AreIsomorphic]

GroupTheory[DihedralGroup]

GroupTheory[GroupOrder]

https://en.wikipedia.org/wiki/Quaternion_group