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.
Observed surface forms (6)
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) ⓘ |
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)