 QuaternionGroup - Maple Help

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GroupTheory

 QuaternionGroup
 construct a generalized quaternion group Calling Sequence QuaternionGroup( n ) QuaternionGroup( f ) Parameters

 n - (optional) an integer greater than or equal to $3$. f - (optional) equation of the form form = "permgroup" (default) or form = "fpgroup" Description

 • The QuaternionGroup( n ) calling sequence constructs a generalized quaternion group of order ${2}^{n}$, where $3\le n$ 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. Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{QuaternionGroup}\left(\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{QuaternionGroup}}{}\left({}\right)$ (1)
 > $\mathrm{QuaternionGroup}\left('\mathrm{form}'="permgroup"\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{QuaternionGroup}}{}\left({}\right)$ (2)
 > $\mathrm{QuaternionGroup}\left('\mathrm{form}'="fpgroup"\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{QuaternionGroup}}{}\left({}\right)$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${8}$ (4)

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

 > $\mathrm{SearchSmallGroups}\left('\mathrm{order}'=8,'\mathrm{abelian}'=\mathrm{false}\right)$
 $\left[{8}{,}{3}\right]{,}\left[{8}{,}{4}\right]$ (5)

One of these is the Quaternion group.

 > $\mathrm{IdentifySmallGroup}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${8}{,}{4}$ (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.

 > $\mathrm{AreIsomorphic}\left(\mathrm{QuaternionGroup}\left(\right),\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (7)

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

 > $\mathrm{ctQ}≔\mathrm{CharacterTable}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{CharacterTable}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (8)
 > $\mathrm{ctD}≔\mathrm{CharacterTable}\left(\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{CharacterTable}}{}\left({\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right)$ (9)
 > $\mathrm{EqualEntries}\left(\mathrm{GetMatrix}\left(\mathrm{ctQ}\right),\mathrm{GetMatrix}\left(\mathrm{ctD}\right)\right)$
 ${\mathrm{true}}$ (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.)

 > $\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$ > $\mathrm{DrawCayleyTable}\left(\mathrm{QuaternionGroup}\left(\right)\right)$ > $\mathrm{DrawSubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right)$ 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.

 > $\mathrm{andmap}\left(\mathrm{IsNormal},\mathrm{convert}\left(\mathrm{SubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right),'\mathrm{list}'\right),\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (11)

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

 > $\mathrm{map}\left(\mathrm{GroupOrder},\left[\mathrm{Centre},\mathrm{DerivedSubgroup},\mathrm{FrattiniSubgroup}\right]\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$
 $\left[{2}{,}{2}{,}{2}\right]$ (12)
 > $\mathrm{map}\left(\mathrm{op}@\mathrm{Generators},\left\{\mathrm{Centre},\mathrm{DerivedSubgroup},\mathrm{FrattiniSubgroup}\right\}\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$
 $\left\{{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{...}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}\right\}$ (13)

The quaternion group of order $8$ is Hamiltonian, but generalized quaternion groups of larger order are not.

 > $\mathrm{IsHamiltonian}\left(\mathrm{QuaternionGroup}\left(\right)\right)$
 ${\mathrm{true}}$ (14)
 > $\mathrm{IsHamiltonian}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$
 ${\mathrm{false}}$ (15)

Quaternion groups do not have perfect order classes.

 > $\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{QuaternionGroup}\left(4\right)\right)$
 ${\mathrm{false}}$ (16) Compatibility

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