Orthogonal group

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In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose.

An important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1. This group is also called the rotation group, because, in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2), SO(3) and SO(4).

The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form[1] on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space F n of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, denoted O(n, F ), is the set of n×n orthogonal matrices with entries from F, with the group operation of matrix multiplication. This is a subgroup of the general linear group GL(n, F ) given by

where QT is the transpose of Q and I is the identity matrix.

This article mainly discusses the orthogonal groups of quadratic forms that may be expressed over some bases as the dot product; over the reals, they are the positive definite quadratic forms. Over the reals, for any non-degenerate quadratic form, there is a basis, on which the matrix of the form is a diagonal matrix such that the diagonal entries are either 1 or −1. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O(p, q), where p is the number of ones and q the number of negative ones. For details, see indefinite orthogonal group.

The derived subgroup Ω(n, F ) of O(n, F) is an often studied object because, when F is a finite field, Ω(n, F ) is often[clarification needed] a central extension of a finite simple group.

Both O(n, F ) and SO(n, F ) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix. The Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.


The determinant of any orthogonal matrix is either 1 or −1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n, F ) known as the special orthogonal group SO(n, F ), consisting of all proper rotations. (More precisely, SO(n, F ) is the kernel of the Dickson invariant, discussed below.). By analogy with GL–SL (general linear group, special linear group), the orthogonal group is sometimes called the general orthogonal group and denoted GO, though this term is also sometimes used for indefinite orthogonal groups O(p, q). The term rotation group can be used to describe either the special or general orthogonal group.

In even and odd dimension[edit]

The structure of the orthogonal group differs in certain respects between even and odd dimensions; for example, over ordered fields (such as R) the I element is orientation-preserving in even dimensions, but orientation-reversing in odd dimensions. When this distinction is to be emphasized, the groups may be denoted O(2k) and O(2k + 1), reserving n for the dimension of the space (n = 2k or n = 2k + 1). The letters p or r are also used, indicating the rank of the corresponding Lie algebra; in odd dimension the corresponding Lie algebra is , while in even dimension the Lie algebra is .

The distinction between even and odd dimensions arises from considering the maximal torus (discussed below). The groups SO(2k) and SO(2k + 1) have the same maximal torus, but different root systems.

Over the real number field[edit]

Over the field R of real numbers, the orthogonal group O(n, R) and the special orthogonal group SO(n, R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n − 1)/2. O(n, R) has two connected components, with SO(n, R) being the identity component, i.e., the connected component containing the identity matrix.

Presentation as real algebraic variety[edit]

If we consider the set of matrices as and let be column vectors representing a matrix, then the orthogonality condition is

Because of the symmetry of these matrices, this gives scalar equations. This proves that O(n) is an algebraic set of dimension

which is a complete intersection.

This algebraic set has two irreducible components, depending on the sign of the determinant (that is det(A) = 1 or det(A) = –1), which are smooth algebraic varieties of the same dimension n(n – 1)/2. The component with det(A) = 1 is SO(n).

Geometric interpretation[edit]

The real orthogonal and real special orthogonal groups have the following geometric interpretations:

O(n, R) is a subgroup of the Euclidean group E(n), the group of isometries of Rn; it contains those that leave the origin fixed – O(n, R) = E(n) ∩ GL(n, R). It is the symmetry group of the sphere (n = 3) or (n − 1)-sphere and all objects with spherical symmetry, if the origin is chosen at the center.

SO(n, R) is a subgroup of E+(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed – SO(n, R) = E+(n) ∩ GL(n, R) = E(n) ∩ GL+(n, R). It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

I} is a normal subgroup and even a characteristic subgroup of O(n, R), and, if n is even, also of SO(n, R). If n is odd, O(n, R) is the internal direct product of SO(n, R) and I}. For every positive integer k the cyclic group Ck of k-fold rotations is a normal subgroup of O(2, R) and SO(2, R).

Relative to suitable orthogonal bases, the isometries are of the form:

where the matrices R1, ..., Rk are 2-by-2 rotation matrices in orthogonal planes of rotation. As a special case, known as Euler's rotation theorem, any (non-identity) element of SO(3, R) is a rotation about a uniquely defined axis.

The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).

The symmetry group of a circle is O(2, R). The orientation-preserving subgroup SO(2, R) is isomorphic (as a real Lie group) to the circle group, also known as U(1). This isomorphism sends the complex number exp(φ i) = cos φ + i sin φ of absolute value 1 to the special orthogonal matrix

The group SO(3, R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering, and there are numerous charts on SO(3).

Maximal tori and Weyl groups[edit]

A maximal torus T for SO(2n), of rank n, is given by the block-diagonal matrices

where the Rj are 2-by-2 rotation matrices. The image T ×{1} of the same torus under the block-diagonal inclusion

is a maximal torus for SO(2n+1). The Weyl group of SO(2n+1) is the semidirect product of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each {±1} factor of {±1}n acts on the corresponding circle factor of T ×{1} by inversion, and the symmetric group Sn acts on both {±1}n and T ×{1} by permuting factors. The elements of the Weyl group are represented by matrices in O(2n) × {±1}. The Sn factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The {±1}n component is represented by block-diagonal matrices with 2-by-2 blocks either

with the last component ±1 chosen to make the determinant 1.

The Weyl group of SO(2n) is the subgroup of that of SO(2n + 1), where Hn−1 < {±1}n is the kernel of the product homomorphism {±1}n → {±1} given by ; that is Hn−1 < {±1}n is the subgroup with an even number of minus signs. The Weyl group of SO(2n) is represented in SO(2n) by the preimages under the standard injection SO(2n) → SO(2n+ 1) of the representatives for the Weyl group of SO(2n + 1). Those matrices with an odd number of blocks have no remaining final −1 coordinate to make their determinants positive, and hence cannot be represented in SO(2n).

Low-dimensional topology[edit]

The low-dimensional (real) orthogonal groups are familiar spaces:

Fundamental group[edit]

In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R) is cyclic of order 2,[3] and the spin group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group Spin(2) is the unique connected 2-fold cover).

Homotopy groups[edit]

Generally, the homotopy groups πk(O) of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. 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. On the other hand, Sn is a homogeneous space for O(n + 1), and one has the following fiber bundle:

which can be understood as "The orthogonal group O(n + 1) acts transitively on the unit sphere Sn, and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower. Thus the natural inclusion O(n) → O(n + 1) is (n − 1)-connected, so the homotopy groups stabilize, and πk(O(n+1)) = πk(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

From Bott periodicity we obtain Ω8OO, therefore the homotopy groups of O are 8-fold periodic, meaning πk + 8(O) = πk(O), and one needs only to list the lower 8 homotopy groups:

Relation to KO-theory[edit]

Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: πk(O) = πk + 1(BO). Setting KO = BO × Z = Ω−1O × Z (to make π0 fit into the periodicity), one obtains:

Computation and interpretation of homotopy groups[edit]

Low-dimensional groups[edit]

The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.

  • π0(O) = π0(O(1)) = Z/2Z, from orientation-preserving/reversing (this class survives to O(2) and hence stably)
  • π1(O) = π1(SO(3)) = Z/2Z, which is spin comes from SO(3) = RP3 = S3/(Z/2Z).
  • π2(O) = π2(SO(3)) = 0, which surjects onto π2(SO(4)); this latter thus vanishes.
Lie groups[edit]

From general facts about Lie groups, π2(G) always vanishes, and π3(G) is free (free abelian).

Vector bundles[edit]

From the vector bundle point of view, π0(KO) is vector bundles over S0, which is two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so π0(KO) = Z is dimension.

Loop spaces[edit]

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. Using π0, O and O/U have two components, KO = BO × Z and KSp = BSp × Z have countably many components, and the rest are connected.

Interpretation of homotopy groups[edit]

In a nutshell:[4]

Let R be any of the four division algebras R, C, H, O, and let LR be the tautological line bundle over the projective line RP1, and [LR] its class in K-theory. Noting that RP1 = S1, CP1 = S2, HP1 = S4, OP1 = S8, these yield vector bundles over the corresponding spheres, and

  • π1(KO) is generated by [LR]
  • π2(KO) is generated by [LC]
  • π4(KO) is generated by [LH]
  • π8(KO) is generated by [LO]

From the point of view of symplectic geometry, π0(KO) ≅ π8(KO) = Z can be interpreted as the Maslov index, thinking of it as the fundamental group π1(U/O) of the stable Lagrangian Grassmannian as U/O ≅ Ω7(KO), so π1(U/O) = π1+7(KO).

Whitehead tower[edit]

The orthogonal group anchors a Whitehead tower:

which is obtained by successively removing (killing) homotopy groups of increasing order. 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. The homotopy groups that are killed are in turn π0(O) to obtain SO from O, π1(O) to obtain Spin from SO, π3(O) to obtain String from Spin, and then π7(O) and so on to obtain the higher order branes.

Over the complex number field[edit]

Over the field C of complex numbers, O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1)/2 over C (it means the dimension over R is twice that). O(n, C) has two connected components, and SO(n, C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact.

Just as in the real case SO(n, C) is not simply connected. For n > 2 the fundamental group of SO(n, C) is cyclic of order 2 whereas the fundamental group of SO(2, C) is infinite cyclic.

Over finite fields[edit]

Orthogonal groups can also be defined over finite fields Fq, where q is a power of a prime p.

Over finite fields of characteristic not equal to 2, orthogonal groups come in two types in even dimension: O+(2n, q) and O(2n, q); and one type in odd dimension: O(2n + 1, q).[5]

If V is the vector space on which the orthogonal group G acts, it can be written as a direct orthogonal sum as follows:

where Li are hyperbolic lines and W contains no singular vectors. If W is the zero subspace, then G is of plus type. If W is one-dimensional then G has odd dimension. If W has dimension 2, G is of minus type.

In the special case where n = 1, Oϵ(2, q) is a dihedral group of order 2(q − ϵ).

We have the following formulas for the order of O(n, q), when the characteristic is not two:

If −1 is a square in Fq

If −1 is a non-square in Fq

The Dickson invariant[edit]

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Z/2Z (integers modulo 2), taking the value 0 in case the element is the product of an even number of reflections, and the value of 1 otherwise.[6]

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). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant. 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[6] and usually has index 2 in O(n, F ).[7] When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n, F ) is commonly defined to be the elements of O(n, F ) with determinant 1. Each element in O(n, F ) has determinant ±1. 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).

Orthogonal groups of characteristic 2[edit]

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.)

  • Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4-dimensional over the field with 2 elements and the Witt index is 2.[8] Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector u takes a vector v to v + B(v, u)/Q(u) · u where B is the bilinear form and Q is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes v to v − 2·B(v, u)/Q(u) · u.
  • The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since I = −I.
  • In odd dimensions 2n + 1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension 2n, acted upon by the orthogonal group.
  • In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.

The spinor norm[edit]

The spinor norm is a homomorphism from an orthogonal group over a field F to the quotient group F×/(F×)2 (the multiplicative group of the field F up to square elements), that takes reflection in a vector of norm n to the image of n in F×/(F×)2.[9]

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.

Galois cohomology and orthogonal groups[edit]

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 H1, 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.

Here μ2 is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from H0(OV), which is simply the group OV(F) of F-valued points, to H12) is essentially the spinor norm, because H12) is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from H1 of the orthogonal group, to the H2 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.

Lie algebra[edit]

The Lie algebra corresponding to Lie groups O(n, F ) and SO(n, F ) consists of the skew-symmetric n × n matrices, with the Lie bracket [ , ] given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by or , and called the orthogonal Lie algebra or special orthogonal Lie algebra. 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.

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.

More intrinsically, given a vector space with an inner product[clarification needed], the special orthogonal Lie algebra is given by the bivectors on the space, which are sums of simple bivectors (2-blades) vw. The correspondence is given by the map where v is the covector dual to the vector v;[clarification needed] in coordinates these are exactly the elementary skew-symmetric[clarification needed] matrices.

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. Generalizing the inner product with a nondegenerate form yields the indefinite orthogonal Lie algebras

Related groups[edit]

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

The inclusions O(n) ⊂ U(n) ⊂ Sp(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.

Lie subgroups[edit]

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:

– preserve an axis
U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; SU(n) also preserves a complex orientation.

Lie supergroups[edit]

The orthogonal group O(n) is also an important subgroup of various Lie groups:

Conformal group[edit]

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 Rn is denoted CO(n) for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If n is odd, these two subgroups do not intersect, and they are a direct product: CO(2k + 1) = O(2k + 1) × R, where R = R∖{0} is the real multiplicative group, while if n is even, these subgroups intersect in ±1, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: CO(2k) = O(2k) × R+.

Similarly one can define CSO(n); note that this is always: CSO(n) = CO(n) ∩ GL+(n) = SO(n) × R+.

Discrete subgroups[edit]

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.[note 2] These subgroups are known as point group and can be realized as the symmetry groups of polytopes. 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.

Other finite subgroups include:

Covering and quotient groups[edit]

The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively:

These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:

Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k + 1) is a 1-to-1 cover, i.e., isomorphic to SO(2k + 1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact special orthogonal Lie algebra, – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.[note 4]

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

Principal homogeneous space: Stiefel manifold[edit]

The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold Vn(Rn) 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.

The other Stiefel manifolds Vk(Rn) for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.

See also[edit]

Specific transforms[edit]

Specific groups[edit]

Related groups[edit]

Lists of groups[edit]


  1. ^ The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other.
  2. ^ Infinite subsets of a compact space have an accumulation point and are not discrete.
  3. ^ O(n) ∩ GL(n, Z) equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be ±1 (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.
  4. ^ In odd dimension, SO(2k + 1) ≅ PSO(2k + 1) is centerless (but not simply connected), while in even dimension SO(2k) is neither centerless nor simply connected.


  1. ^ For base fields of characteristic not 2, the definition in terms of a symmetric bilinear form is equivalent to that in terms of a quadratic form, but in characteristic 2 these notions differ.
  2. ^ Hall 2015 Section 1.3.4
  3. ^ Hall 2015 Proposition 13.10
  4. ^ John Baez "This Week's Finds in Mathematical Physics" week 105
  5. ^ Wilson, Robert A. (2009). The finite simple groups. Graduate Texts in Mathematics. 251. London: Springer. pp. 69–75. ISBN 978-1-84800-987-5. Zbl 1203.20012. 
  6. ^ a b Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, 294, Berlin etc.: Springer-Verlag, p. 224, ISBN 3-540-52117-8, Zbl 0756.11008 
  7. ^ (Taylor 1992, page 160)
  8. ^ (Grove 2002, Theorem 6.6 and 14.16)
  9. ^ Cassels 1978, p. 178


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