affine group of R^n
E518474
The affine group of ℝⁿ is the group of all invertible affine transformations of n-dimensional real space, combining linear transformations with translations.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical group
ⓘ
matrix group ⓘ transformation group ⓘ |
| actsOn | R^n ⓘ |
| alsoKnownAs |
Aff(R^n)
NERFINISHED
ⓘ
general affine group of R^n NERFINISHED ⓘ |
| contains |
GL(n,R)
NERFINISHED
ⓘ
group of translations of R^n ⓘ |
| containsAsSubgroup |
Euclidean group E(n)
NERFINISHED
ⓘ
orthogonal group O(n) NERFINISHED ⓘ special orthogonal group SO(n) NERFINISHED ⓘ |
| embedsInto | GL(n+1,R) NERFINISHED ⓘ |
| hasActionType | affine action ⓘ |
| hasComponent |
A ∈ GL(n,R)
ⓘ
b ∈ R^n ⓘ |
| hasDimension | n^2 + n ⓘ |
| hasElementForm | x ↦ A x + b ⓘ |
| hasIdentityElement | x ↦ x ⓘ |
| hasInverseFormula | (A,b)^{-1} = (A^{-1}, -A^{-1} b) ⓘ |
| hasLieAlgebra | affine Lie algebra of R^n ⓘ |
| hasLieAlgebraStructure | R^n ⋊ gl(n,R) ⓘ |
| hasMultiplicationRule | (A,b)(C,d) = (AC, A d + b) ⓘ |
| hasNormalSubgroup | translation subgroup R^n ⓘ |
| hasQuotientBy | (affine group of R^n, translation subgroup R^n) ≅ GL(n,R) ⓘ |
| hasStabilizerOfPoint | GL(n,R) NERFINISHED ⓘ |
| hasStructure | R^n ⋊ GL(n,R) ⓘ |
| hasTopology | standard Lie group topology from R^{n^2+n} ⓘ |
| hasTrivialCenterFor | n ≥ 2 ⓘ |
| isConnected | true ⓘ |
| isDefinedOver | real numbers R ⓘ |
| isFaithfulActionOn | R^n ⓘ |
| isGeneratedBy |
linear transformations in GL(n,R)
ⓘ
translations of R^n ⓘ |
| isGroupUnder | composition of maps ⓘ |
| isLinearGroupVia | (A,b) ↦ [[A,b],[0,1]] ⓘ |
| isMaximalGroupPreserving | affine lines in R^n ⓘ |
| isNonAbelianFor | n ≥ 1 ⓘ |
| isRealLieGroup | true ⓘ |
| isSemidirectProductOf |
GL(n,R)
NERFINISHED
ⓘ
R^n ⓘ |
| isSubgroupOf |
diffeomorphism group of R^n
ⓘ
homeomorphism group of R^n ⓘ |
| isUsedIn |
affine geometry
ⓘ
differential geometry ⓘ geometric group theory ⓘ |
| preserves | affine structure of R^n ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.