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It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a subgroup, called the special orthogonal group , and denoted SO n.

It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group , generalizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point in dimension 2 or a line in dimension 3.

In the other connected component all orthogonal matrices have —1 as a determinant. More generally, given a non-degenerate symmetric bilinear form or quadratic form [1] on a vector space over a field , the orthogonal group of the form is the group of invertible linear maps that preserve the form.

The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product , or, equivalently, the quadratic form is the sum of the square of the coordinates.

All orthogonal groups are algebraic groups , since the condition of preserving a form can be expressed as an equality of matrices. The name of "orthogonal group" originates from the following characterization of its elements.

Given a Euclidean vector space E of dimension n , the elements of the orthogonal group O n are, up to a uniform scaling homothecy , the linear maps from E to E that map orthogonal vectors to orthogonal vectors.

Let E n be the group of the Euclidean isometries of a Euclidean space S of dimension n. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic.

This stabilizer is or, more exactly, is isomorphic to O n , since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.

There is a natural group homomorphism p from E n to O n , which is defined by. The kernel of p is the vector space of the translations.

So, the translation form a normal subgroup of E n , the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to O n.

Moreover, the Euclidean group is a semidirect product of O n and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of O n.

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.

The orthogonal matrices with determinant 1 form a subgroup called the special orthogonal group , denoted SO n , consisting of all direct isometries of O n , which are those that preserve the orientation of the space.

For every positive integer k the cyclic group C k of k -fold rotations is a normal subgroup of O 2 and SO 2.

For any element of O n there is an orthogonal basis, where its matrix has the form. This results from the spectral theorem by regrouping eigenvalues that are complex conjugate , and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to 1.

The element belongs to SO n if and only if there are an even number of —1 on the diagonal. Reflections are the elements of O n whose canonical form is.

In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.

In dimension two, every rotation is the product of two reflections. Every element of O n is the product of at most n reflections. This results immediately from the above canonical form and the case of dimension two.

The symmetry group of a circle is O 2. The orientation-preserving subgroup SO 2 is isomorphic as a real Lie group to the circle group , also known as U 1 , the multiplicative group of the complex numbers of absolute value equal to one.

In higher dimension, O n has a more complicated structure in particular, it is no longer commutative. The topological structures of the n -sphere and O n are strongly correlated, and this correlation is widely used for studying both topological spaces.

The group O n has two connected components , with SO n being the identity component , that is, the connected component containing the identity matrix.

This proves that O n is an algebraic set. Moreover, it can be proved that its dimension is. This implies that all its irreducible components have the same dimension, and that it has no embedded component.

In O 2 n and SO 2 n , for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form.

The S n factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The low-dimensional real orthogonal groups are familiar spaces :.

However, one can compute the homotopy groups of the stable orthogonal group aka the infinite orthogonal group , defined as the direct limit of the sequence of inclusions:.

Since the inclusions are all closed, hence cofibrations , this can also be interpreted as a union.

The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups. Using concrete descriptions of the loop spaces in Bott periodicity , one can interpret the higher homotopies of O in terms of simpler-to-analyze homotopies of lower order.

In a nutshell: [5]. The orthogonal group anchors a Whitehead tower :. This is done by constructing short exact sequences starting with an Eilenberg—MacLane space for the homotopy group to be removed.

The first few entries in the tower are the spin group and the string group , and are preceded by the fivebrane group.

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. So, in the remainder of this section, it is supposed that neither p nor q is zero.

The subgroup of the matrices of determinant 1 in O p , q is denoted SO p , q. The group O p , q has four connected components, depending on whether an element preserves orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite.

The group O 3, 1 is the Lorentz group that is fundamental in relativity theory. Here the 3 corresponds to space coordinates, and 1 corresponds to the time.

Over the field C of complex numbers , every non-degenerate quadratic form is a sum of squares. There is thus only one orthogonal group for each dimension over the complexes, that is usually denoted O n , C.

It can be identified with the group of complex orthogonal matrices , that is the complex matrices whose product with their transpose is the identity matrix.

Similarly as in the real case, O n , C has two connected components. The component of the identity consists of all matrices of O n , C with 1 as their determinant, and is denoted SO n , C.

Just as in the real case SO n , C is not simply connected. Over a field of characteristic different from two, two quadratic forms are equivalent if their matrices are congruent , that is if a change of basis transforms the matrix of the first form into the matrix of the second form.

Two equivalent quadratic forms have clearly the same orthogonal group. The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension.

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.

Chevalley—Warning theorem asserts that over a finite field the dimension of W is at most two. This implies that if the dimension of V is even, there are only two orthogonal groups, depending whether the dimension of W zero or two.

In the case of O — 2 n , q , the above x and y are conjugate , and are therefore the image of each other by the Frobenius automorphism.

When the characteristic is not two, the order of the orthogonal groups are [7]. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.

The special orthogonal group is the kernel of the Dickson invariant [8] and usually has index 2 in O n , F. Thus in characteristic 2, the determinant is always 1.

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Over a field of characteristic different from two, two quadratic forms are equivalent if their matrices are congruent , that is if a change of basis transforms the matrix of the first form into the matrix of the second form.

Two equivalent quadratic forms have clearly the same orthogonal group. The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension.

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.

Chevalley—Warning theorem asserts that over a finite field the dimension of W is at most two. This implies that if the dimension of V is even, there are only two orthogonal groups, depending whether the dimension of W zero or two.

In the case of O — 2 n , q , the above x and y are conjugate , and are therefore the image of each other by the Frobenius automorphism.

When the characteristic is not two, the order of the orthogonal groups are [7]. Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.

The special orthogonal group is the kernel of the Dickson invariant [8] and usually has index 2 in O n , F.

Thus in characteristic 2, the determinant is always 1. The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way in all dimensions.

Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. Formerly these groups were known as the hypoabelian groups , but this term is no longer used.

For the usual orthogonal group over the reals, it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.

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 phenomena is concerned.

The first point is that quadratic forms over a field can be identified as a Galois H 1 , or twisted forms torsors of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.

The 'spin' name of the spinor norm can be explained by a connection to the spin group more accurately a pin group. This may now be explained quickly by Galois cohomology which however postdates the introduction of the term by more direct use of Clifford algebras.

The spin covering of the orthogonal group provides a short exact sequence of algebraic groups. There is also the connecting homomorphism from H 1 of the orthogonal group, to the H 2 of the kernel of the spin covering.

The cohomology is non-abelian so that this is as far as we can go, at least with the conventional definitions. One Lie algebra corresponds to both groups.

Since the group SO n is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to ordinary representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups.

The projective representations of SO n are just linear representations of the universal cover, the spin group Spin n. The latter are the so-called spin representation , which are important in physics.

Over a field of characteristic 2 we consider instead the alternating endomorphisms. The correspondence is given by:. Over real numbers, this characterization is used in interpreting the curl of a vector field naturally a 2-vector as an infinitesimal rotation or "curl", hence the name.

The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups.

These are listed below. In physics, particularly in the areas of Kaluza—Klein compactification, it is important to find out the subgroups of the orthogonal group.

The main ones are:. The orthogonal group O n is also an important subgroup of various Lie groups:. Being isometries , real orthogonal transforms preserve angles , and are thus conformal maps , though not all conformal linear transforms are orthogonal.

In classical terms this is the difference between congruence and similarity , as exemplified by SSS side-side-side congruence of triangles and AAA angle-angle-angle similarity of triangles.

The group of conformal linear maps of R n is denoted CO n for the conformal orthogonal group , and consists of the product of the orthogonal group with the group of dilations.

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups. A very important class of examples are the finite Coxeter groups , which include the symmetry groups of regular polytopes.

Dimension 3 is particularly studied — see point groups in three dimensions , polyhedral groups , and list of spherical symmetry groups.

In 2 dimensions, the finite groups are either cyclic or dihedral — see point groups in two dimensions. The orthogonal group is neither simply connected nor centerless , and thus has both a covering group and a quotient group , respectively:.

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.

The principal homogeneous space for the orthogonal group O n is the Stiefel manifold V n R n of orthonormal bases orthonormal n -frames. In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group.

Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

From Wikipedia, the free encyclopedia. Group of isometries of a Euclidean vector space or, more generally, of a vector space equipped with a quadratic form.

Basic notions. Subgroup Normal subgroup Quotient group Semi- direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics.

Finite groups. Discrete groups Lattices. Topological and Lie groups. Algebraic groups. Linear algebraic group Reductive group Abelian variety Elliptic curve.

This section may be confusing or unclear to readers. In particular, most notations are undefined; no context for explaining why these consideration belong to the article.

Moreover, the section consists esentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them no reference, no link to articles about the methods of computation that are used, no sketch of proofs.

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Main article: Indefinite orthogonal group. The comparison of this proof with the real case may be illuminating. Main article: Conformal group.

Main article: Stiefel manifold. The finite simple groups. Graduate Texts in Mathematics. London: Springer. Cassels, J. Categories : Lie groups Quadratic forms Euclidean symmetries Linear algebraic groups.

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