construct a generalized quaternion group
QuaternionGroup( n )
QuaternionGroup( f )
(optional) an integer greater than or equal to 3.
(optional) equation of the form form = "permgroup" (default) or form = "fpgroup"
The QuaternionGroup( n ) calling sequence constructs a generalized quaternion group of order 2n, where 3≤n is an integer.
The argument n is optional, and is taken to be equal to 3 by default, so the calling sequence QuaternionGroup() returns a quaternion group of order 8.
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 generalized quaternion group is constructed as a permutation group by default.
But, 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.
There are only two non-Abelian groups of order eight.
One of these is the Quaternion group.
The dihedral group of order 8 (and degree 4) is the other group of order 8. It is not isomorphic to the quaternion group.
However, the quaternion and dihedral groups of order eight do have the same character tables.
ctQ ≔ CharacterTable⁡QuaternionGroup⁡
ctD ≔ CharacterTable⁡DihedralGroup⁡4
(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.)
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.
Like the dihedral group of order 8, the quaternion group is an extra-special 2-group.
The quaternion group of order 8 is Hamiltonian, but generalized quaternion groups of larger order are not.
Quaternion groups do not have perfect order classes.
The GroupTheory[QuaternionGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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