The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n).
By extension, for any field F, an n × n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n × n orthogonal matrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is
This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by g (for details, see Affine space § Subtraction and Weyl's axioms).
By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices, which are the matrices such that It follows from this equation that the square of the determinant of Q equals 1, and thus the determinant of Q is either 1 or −1.
This implies that the orthogonal group is an internal semidirect product of SO(n) and any subgroup formed with the identity and a reflection.
The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.
In O(2n + 1) and SO(2n + 1), the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal.
blocks have no remaining final −1 coordinate to make their determinants positive, and hence cannot be represented in SO(2n).
This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed.
Over the real numbers, nondegenerate quadratic forms are classified by Sylvester's law of inertia, which asserts that, on a vector space of dimension n, such a form can be written as the difference of a sum of p squares and a sum of q squares, with p + q = n. In other words, there is a basis on which the matrix of the quadratic form is a diagonal matrix, with p entries equal to 1, and q entries equal to −1.
The pair (p, q) called the inertia, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix.
The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted O(p, q).
Over the field C of complex numbers, every non-degenerate quadratic form in n variables is equivalent to x12 + ... + xn2.
Thus, up to isomorphism, there is only one non-degenerate complex quadratic space of dimension n, and one associated orthogonal group, usually denoted O(n, C).
More precisely, Witt's decomposition theorem asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form Q can be decomposed as a direct sum of pairwise orthogonal subspaces where each Li is a hyperbolic plane (that is there is a basis such that the matrix of the restriction of Q to Li has the form
It results that there is only one orthogonal group that is denoted O(2n + 1, q), where q is the number of elements of the finite field (a power of an odd prime).
For studying the orthogonal group of Oε(2, q), one can suppose that the matrix of the quadratic form is
For further studying the orthogonal group, it is convenient to introduce a square root α of ω.
In the real case, the corresponding isomorphisms are: where C is the circle of the complex numbers of norm one; with
[8] Algebraically, the Dickson invariant can be defined as D(f) = rank(I − f) modulo 2, where I is the identity (Taylor 1992, Theorem 11.43).
The special orthogonal group is the kernel of the Dickson invariant[8] and usually has index 2 in O(n, F ).
Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section.
In the theory of Galois cohomology of algebraic groups, some further points of view are introduced.
They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomenon is concerned.
This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras).
Over real numbers, these Lie algebras for different n are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension Bk, where n = 2k + 1, while in even dimension Dr, where n = 2r.
More generally, given a vector space V (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form (⋅, ⋅), the special orthogonal Lie algebra consists of tracefree endomorphisms
The correspondence is given by: This description applies equally for the indefinite special orthogonal Lie algebras
The inclusions O(n) ⊂ U(n) ⊂ USp(2n) and USp(n) ⊂ U(n) ⊂ O(2n) are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, U(n)/O(n) is the Lagrangian Grassmannian.
In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group.