E(n)
E121355
E(n) is the Euclidean group of dimension n, consisting of all distance-preserving transformations (rotations, reflections, and translations) of n-dimensional Euclidean space.
All labels observed (1)
| Label | Occurrences |
|---|---|
| E(n) canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1056978 — 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: E(n) Context triple: [Euclidean space, hasIsometryGroup, E(n)]
-
A.
N
The N is a New York City Subway service that runs along the Broadway Line in Manhattan and connects Queens, Manhattan, and Brooklyn.
-
B.
Enz
The Enz is a river in southwestern Germany that flows through the Black Forest region before joining the Neckar River.
-
C.
NE
NE is the common abbreviation for the New England Revolution, a professional Major League Soccer club based in the Greater Boston area.
-
D.
NE
NE is the official two-letter United States Postal Service abbreviation used to designate the state of Nebraska.
-
E.
E.I.
"E.I." is a hit hip-hop single by American rapper Nelly from his debut album "Country Grammar."
- 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: E(n) Target entity description: E(n) is the Euclidean group of dimension n, consisting of all distance-preserving transformations (rotations, reflections, and translations) of n-dimensional Euclidean space.
-
A.
N
The N is a New York City Subway service that runs along the Broadway Line in Manhattan and connects Queens, Manhattan, and Brooklyn.
-
B.
Enz
The Enz is a river in southwestern Germany that flows through the Black Forest region before joining the Neckar River.
-
C.
NE
NE is the official two-letter United States Postal Service abbreviation used to designate the state of Nebraska.
-
D.
NE
NE is the common abbreviation for the New England Revolution, a professional Major League Soccer club based in the Greater Boston area.
-
E.
E.I.
"E.I." is a hit hip-hop single by American rapper Nelly from his debut album "Country Grammar."
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
isometry group ⓘ mathematical group ⓘ topological group ⓘ |
| actsOn | n-dimensional Euclidean space ⓘ |
| alsoKnownAs |
Euclidean group of dimension n
ⓘ
linked to:
Euclidean group
group of Euclidean isometries of R^n ⓘ |
| canBeRepresentedBy | (A,b) with A in O(n) and b in R^n ⓘ |
| containsSubgroup |
orthogonal group O(n)
ⓘ
special orthogonal group SO(n) ⓘ translation group of R^n ⓘ |
| generalizes |
Euclidean group E(2)
ⓘ
linked to:
Euclidean group
Euclidean group E(3) ⓘ
linked to:
Euclidean group
|
| hasActionProperty | acts transitively on R^n ⓘ |
| hasConnectedComponentOfIdentity |
orientation-preserving Euclidean group E^+(n)
ⓘ
linked to:
Euclidean group
|
| hasDimension | n(n+1)/2 as a Lie group ⓘ |
| hasElementType |
distance-preserving transformation of R^n
ⓘ
glide reflection ⓘ reflection ⓘ rotation ⓘ rotoreflection ⓘ translation ⓘ |
| hasGroupOperation | composition of transformations ⓘ |
| hasIdentityComponent | E^+(n) ⓘ |
| hasIdentityElement | identity isometry of R^n ⓘ |
| hasInverseOperation | inverse isometry ⓘ |
| hasLieAlgebra | e(n) ⓘ |
| hasLieAlgebraDimension | n(n+1)/2 ⓘ |
| hasNormalSubgroup | translation group R^n ⓘ |
| hasQuotientByNormalSubgroup | O(n) ⓘ |
| hasSemidirectProductDecomposition | R^n ⋊ O(n) ⓘ |
| hasStabilizerOfPoint | O(n) ⓘ |
| hasTopology | subspace topology from GL(n+1,R) via matrix representation ⓘ |
| isClosedSubgroupOf | affine group of R^n ⓘ |
| isHausdorff | true ⓘ |
| isHomogeneousSpace | R^n ≅ E(n)/O(n) ⓘ |
| isIsometryGroupOf | R^n ⓘ |
| isLocallyCompact | true ⓘ |
| isNonabelian | true ⓘ |
| isNoncompact | true ⓘ |
| isSecondCountable | true ⓘ |
| isSemidirectProductOf |
O(n)
ⓘ
R^n ⓘ |
| isSubgroupOf | affine group of R^n ⓘ |
| isUsedIn |
classical mechanics
ⓘ
geometry of Euclidean spaces ⓘ rigid body kinematics ⓘ |
| preserves |
Euclidean distance
ⓘ
Euclidean metric ⓘ |
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: E(n) Description of subject: E(n) is the Euclidean group of dimension n, consisting of all distance-preserving transformations (rotations, reflections, and translations) of n-dimensional Euclidean space.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.