|
gptkbp:instanceOf
|
gptkb:group_of_people
|
|
gptkbp:alsoKnownAs
|
gptkb:Q8
|
|
gptkbp:automorphismGroup
|
gptkb:S_4
|
|
gptkbp:centralTo
|
{1, -1}
|
|
gptkbp:containsElement
|
{1, -1, i, -i, j, -j, k, -k}
|
|
gptkbp:generation
|
i
j
|
|
gptkbp:hasClassEquation
|
8 = 1 + 1 + 2 + 2 + 2
|
|
gptkbp:hasConjugacyClasses
|
5
|
|
gptkbp:hasElementOrder
|
2
1
4
|
|
gptkbp:hasExponent
|
4
|
|
gptkbp:hasNormalSubgroups
|
all subgroups are normal
|
|
gptkbp:hasSubgroup
|
gptkb:Klein_four-group
quaternions
cyclic group of order 4
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
quaternion group
|
|
gptkbp:isDirectProductOf
|
no nontrivial direct product decomposition
|
|
gptkbp:isGroupOfOrder
|
8
|
|
gptkbp:isHamiltonian
|
true
|
|
gptkbp:isHamiltonianGroup
|
true
|
|
gptkbp:isNilpotent
|
true
|
|
gptkbp:isNonAbelian
|
true
|
|
gptkbp:isNonAbelianGroupOfOrder8
|
true
|
|
gptkbp:isNonCyclic
|
true
|
|
gptkbp:isNonIsomorphicTo
|
gptkb:dihedral_group_of_order_8
|
|
gptkbp:isNotAbelian
|
true
|
|
gptkbp:isNotCyclic
|
true
|
|
gptkbp:isNotDihedral
|
true
|
|
gptkbp:isomorphicTo
|
group of unit quaternions {±1, ±i, ±j, ±k}
|
|
gptkbp:isPerfect
|
false
|
|
gptkbp:isSimple
|
false
|
|
gptkbp:isSmallestHamiltonianGroup
|
true
|
|
gptkbp:isSmallestNonAbelianGroupWithAllSubgroupsNormal
|
true
|
|
gptkbp:isSolvable
|
true
|
|
gptkbp:isUsedAsExampleIn
|
group theory
abstract algebra textbooks
|
|
gptkbp:order
|
8
|
|
gptkbp:presentedBy
|
<i, j | i^4 = 1, i^2 = j^2, ij = ji^{-1}>
|
|
gptkbp:usedIn
|
gptkb:topology
abstract algebra
group theory
quantum mechanics
representation theory
|
|
gptkbp:bfsParent
|
gptkb:Hamiltonian_group
|
|
gptkbp:bfsLayer
|
5
|