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.

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All labels observed (6)

Label	Occurrences
SO(2)	5
special orthogonal group SO(n) canonical	5
SO(n)	3
special orthogonal group	2
SO(d+1)	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 ⓘ linked to: special orthogonal group SO(n)
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) ⓘ linked to: general linear group GL(n,R)
isCompact	true ⓘ
isConnected	true for n ≥ 2 ⓘ
isDoubleCoveredBy	SU(2) ⓘ linked to: rotation group SU(2)
isIsomorphicTo	U(1) ⓘ circle group S¹ ⓘ projective special unitary group PSU(2) ⓘ {1} ⓘ
isKernelOf	determinant map from O(n) to {±1} ⓘ
isLocallyIsomorphicTo	SU(2) × SU(2) ⓘ
isMaximalCompactSubgroupOf	SL(n,ℝ) ⓘ linked to: special linear group SL(n,R)
isNonAbelian	true for n ≥ 3 ⓘ
isSimple	false for n = 2,3,4 ⓘ true for n ≥ 5 ⓘ
isSubsetOf	GL(n,ℝ) ⓘ linked to: general linear group GL(n,R) 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