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orthogonal group

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orthogonalgroup (pluralorthogonal groups)

1. (group theory) For givenn andfieldF (especially whereF is thereal numbers), the group ofn ×northogonal matrices with elements inF, where the group operation is matrix multiplication.
   • 1998,Robert L. Griess, Jr.,Twelve Sporadic Groups, Springer,page 4:

     The symbolO^ε(n,q) fororthogonal groups has been well established in finite group theory as and, throughout the mathematics community,O(n,K) stands for anorthogonal group whenK is the real or complex field.

   • 1999, Gunter Malle, B.H. Matzat,Inverse Galois Theory, Springer,page146:

     Theorem 7.4.Let n ≥ 1. For odd primes${\displaystyle p\not\equiv \pm 1(\mathrm{mod}24)}$ the odd-dimensionalorthogonal groups${\displaystyle O_{2n+1}(p)}$ possess GA-realizations over${\displaystyle \mathbb{Q}}$.

   • 2007, Marcelo Epstein, Marek Elzanowski,Material Inhomogeneities and their Evolution: A Geometric Approach, Springer,page106:

     The normalizer of the fullorthogonal group within the general linear group can be shown to consist of all (commutative) products of spherical dilatations and orthogonal transformations.

DenotedO(n) in the real number case;O(n,F) in the general case.

In the case thatF is thereal numbers, theorthogonal group is equivalently definable as thegroup ofdistance-preservingtransformations of ann-dimensionalEuclidean space that preserve a given fixed point, where the groupoperation is that ofcomposition of transformations.

• indefinite orthogonal group
• special orthogonal group

• symplectic group
• unitary group

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