special orthogonal group SO(n)
E524430
mathematical group
matrix group
real algebraic group
rotation group in 3 dimensions
topological group
trivial group
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
All labels observed (3)
| Label | Occurrences |
|---|---|
| special orthogonal group SO(n) canonical | 2 |
| SO(2) | 1 |
| special orthogonal group SO(2,d) | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425649 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: special orthogonal group SO(n) Context triple: [E(n), containsSubgroup, special orthogonal group SO(n)]
-
A.
orthogonal group O(n)
The orthogonal group O(n) is the group of all n×n real matrices that preserve the standard Euclidean inner product, representing rotations and reflections in n-dimensional space.
-
B.
rotation group SO(3)
The rotation group SO(3) is the group of all rotations in three-dimensional space, represented by 3×3 orthogonal matrices with determinant 1, and plays a central role in classical mechanics, quantum mechanics, and geometry.
-
C.
Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
-
D.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
E.
rotation group SU(2)
The rotation group SU(2) is the Lie group of 2×2 unitary matrices with determinant 1 that serves as the double cover of the three-dimensional rotation group SO(3) and underlies the quantum theory of angular momentum and spin.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: special orthogonal group SO(n) Target entity description: The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
-
A.
orthogonal group O(n)
The orthogonal group O(n) is the group of all n×n real matrices that preserve the standard Euclidean inner product, representing rotations and reflections in n-dimensional space.
-
B.
rotation group SO(3)
The rotation group SO(3) is the group of all rotations in three-dimensional space, represented by 3×3 orthogonal matrices with determinant 1, and plays a central role in classical mechanics, quantum mechanics, and geometry.
-
C.
Lorentz group
The Lorentz group is the mathematical group of spacetime symmetries in special relativity, consisting of all rotations and boosts that preserve the Minkowski spacetime interval.
-
D.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
E.
rotation group SU(2)
The rotation group SU(2) is the Lie group of 2×2 unitary matrices with determinant 1 that serves as the double cover of the three-dimensional rotation group SO(3) and underlies the quantum theory of angular momentum and spin.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical group
ⓘ
matrix group ⓘ real algebraic group ⓘ rotation group in 3 dimensions ⓘ topological group ⓘ trivial group ⓘ |
| actsOn | ℝⁿ by linear isometries ⓘ |
| consistsOf | n×n real matrices ⓘ |
| definedOver | real numbers ⓘ |
| dimension |
3
ⓘ
6 ⓘ n(n−1)/2 as real manifold ⓘ |
| fullName | special orthogonal group NERFINISHED ⓘ |
| fundamentalGroup |
ℤ for n = 2
ⓘ
ℤ/2ℤ ⓘ ℤ/2ℤ for n ≥ 3 ⓘ |
| groupOperation | matrix multiplication ⓘ |
| hasProperty |
determinant 1
ⓘ
orthogonal matrices ⓘ |
| hasTwoComponentsIn | O(n) with O(n) = SO(n) ⊔ (reflection coset) ⓘ |
| identityElement | n×n identity matrix ⓘ |
| inverseOperation | matrix transpose ⓘ |
| isAbelian |
true
ⓘ
true for n ≤ 2 ⓘ |
| isClosedSubgroupOf | GL(n,ℝ) NERFINISHED ⓘ |
| isCompact | true ⓘ |
| isConnected | true for n ≥ 2 ⓘ |
| isDoubleCoveredBy | SU(2) NERFINISHED ⓘ |
| isIsomorphicTo |
U(1)
NERFINISHED
ⓘ
circle group S¹ ⓘ projective special unitary group PSU(2) NERFINISHED ⓘ {1} ⓘ |
| isKernelOf | determinant map from O(n) to {±1} ⓘ |
| isLocallyIsomorphicTo | SU(2) × SU(2) ⓘ |
| isMaximalCompactSubgroupOf | SL(n,ℝ) NERFINISHED ⓘ |
| isNonAbelian | true for n ≥ 3 ⓘ |
| isSimple |
false for n = 2,3,4
ⓘ
true for n ≥ 5 ⓘ |
| isSubsetOf |
GL(n,ℝ)
NERFINISHED
ⓘ
O(n) ⓘ |
| isZariskiClosed | true in Mₙ(ℝ) ⓘ |
| LieAlgebra | so(n) ⓘ |
| LieAlgebraDescription | skew-symmetric n×n real matrices ⓘ |
| preserves |
orientation of ℝⁿ
ⓘ
standard Euclidean inner product on ℝⁿ ⓘ |
| rank | ⌊n/2⌋ ⓘ |
| represents |
orientation-preserving isometries of ℝⁿ fixing the origin
ⓘ
rotations of n-dimensional Euclidean space ⓘ |
| symbol | SO(n) ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: special orthogonal group SO(n) Description of subject: The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
special orthogonal group SO(2,d)
subject surface form:
SO(3)
this entity surface form:
SO(2)