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.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Aff(R^n) | 1 |
| affine group of R^n canonical | 1 |
| general affine group of R^n | 1 |
| special affine group | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5425647 — 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: affine group of R^n Context triple: [E(n), isSubgroupOf, affine group of R^n]
-
A.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
B.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
-
C.
Galilean group
The Galilean group is the mathematical group of spacetime transformations—comprising translations, rotations, and Galilean boosts—that characterize the symmetries of classical Newtonian mechanics.
-
D.
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.
-
E.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
- 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: affine group of R^n Target entity description: The affine group of ℝⁿ is the group of all invertible affine transformations of n-dimensional real space, combining linear transformations with translations.
-
A.
Euclidean group
The Euclidean group is the group of all distance-preserving transformations of Euclidean space, consisting of rotations, reflections, and translations.
-
B.
Lie group
A Lie group is a mathematical structure that is both a smooth manifold and a group, where the group operations are differentiable and used to study continuous symmetries.
-
C.
Galilean group
The Galilean group is the mathematical group of spacetime transformations—comprising translations, rotations, and Galilean boosts—that characterize the symmetries of classical Newtonian mechanics.
-
D.
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.
-
E.
Poincaré group
The Poincaré group is the fundamental symmetry group of special relativity, combining spacetime translations with Lorentz transformations in four-dimensional Minkowski space.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical group
ⓘ
matrix group ⓘ transformation group ⓘ |
| actsOn | R^n ⓘ |
| alsoKnownAs |
Aff(R^n)
ⓘ
linked to:
affine group of R^n
general affine group of R^n ⓘ
linked to:
affine group of R^n
|
| contains |
GL(n,R)
ⓘ
linked to:
general linear group GL(n,R)
group of translations of R^n ⓘ |
| containsAsSubgroup |
Euclidean group E(n)
ⓘ
linked to:
Euclidean group
orthogonal group O(n) ⓘ special orthogonal group SO(n) ⓘ |
| embedsInto |
GL(n+1,R)
ⓘ
linked to:
general linear group GL(n,R)
|
| 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)
ⓘ
linked to:
general linear group GL(n,R)
|
| 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)
ⓘ
linked to:
general linear group GL(n,R)
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 ⓘ |
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: affine group of R^n Description of subject: The affine group of ℝⁿ is the group of all invertible affine transformations of n-dimensional real space, combining linear transformations with translations.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: affine group of R^n
linked to: affine group of R^n
linked to: affine group of R^n