orthogonal group

English

Noun

orthogonal group (plural orthogonal groups)

1. (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 O^ε(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 $p\not \equiv \pm 1{\pmod {24}}$ the odd-dimensional orthogonal groups $O_{2n+1}(p)$ possess GA-realizations over $\mathbb {Q}$.

   • 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

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.

Derived terms

• indefinite orthogonal group
• special orthogonal group

See also

• symplectic group
• unitary group