Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: orthogonal group O(n) | Statement: [affine group of R^n, containsAsSubgroup, orthogonal group O(n)]

Target entity: orthogonal group O(n)
Context triple: [affine group of R^n, containsAsSubgroup, orthogonal group O(n)]

• A. orthogonal group O(n) chosen
  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. special orthogonal group SO(n)
  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.
• C. orthogonal group O(n+1,2)
  The orthogonal group O(n+1,2) is the Lie group of linear transformations preserving a nondegenerate quadratic form of signature (n+1,2), playing a central role in conformal and Lie sphere geometry.
• D. general linear group GL(n,R)
  The general linear group GL(n,ℝ) is the Lie group consisting of all invertible n×n real matrices under matrix multiplication, fundamental in linear algebra and differential geometry.
• E. general linear group GL(n,C)
  The general linear group GL(n,ℂ) is the Lie group consisting of all invertible n×n complex matrices under matrix multiplication, fundamental in linear algebra and representation theory.
• F. None of above.
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target predicate: containsAsSubgroup
Context triple: [affine group of R^n, containsAsSubgroup, orthogonal group O(n)]

• A. hasSubgroupIsomorphicTo chosen
  Indicates that one group contains a subgroup that is structurally identical (isomorphic) to another specified group.
• B. containsGroup
  Indicates that one entity includes or encompasses a specific group of entities within it.
• C. isClosedSubgroupOf
  Indicates that one group is a subgroup of another and is closed in the topological sense within that larger group.
• D. revealedAsSubgroupOf
  Indicates that one group has been newly identified or exposed as being a subgroup or subset of another group.
• E. haveSubgroups
  Indicates that an entity is organized into smaller constituent groups that are part of it.
• F. None of above.