orthogonal group
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English
[edit]Noun
[edit]orthogonal group (plural orthogonal groups)
- (group theory) For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication.
- 1998, Robert L. Griess, Jr., Twelve Sporadic Groups, Springer, page 4:
- The symbol Oε(n,q) for orthogonal groups has been well established in finite group theory as and, throughout the mathematics community, O(n, K) stands for an orthogonal group when K is the real or complex field.
- 1999, Gunter Malle, B.H. Matzat, Inverse Galois Theory, Springer, page 146:
- Theorem 7.4. Let n ≥ 1. For odd primes the odd-dimensional orthogonal groups possess GA-realizations over .
- 2007, Marcelo Epstein, Marek Elzanowski, Material Inhomogeneities and their Evolution: A Geometric Approach, Springer, page 106:
- The normalizer of the full orthogonal group within the general linear group can be shown to consist of all (commutative) products of spherical dilatations and orthogonal transformations.
Usage notes
[edit]Denoted O(n) in the real number case; O(n, F) in the general case.
In the case that F is the real numbers, the orthogonal group is equivalently definable as the group of distance-preserving transformations of an n-dimensional Euclidean space that preserve a given fixed point, where the group operation is that of composition of transformations.