TRANSLATIONS AND THE EUCLIDEAN GROUP

Lemma 5.1. If α: Rn → Rn is an isometry such that α(0) = 0, then α is linear.

Hence the isometries of Rn that fix the origin are precisely the linear isometries. An n × n matrix A is called orthogonal if it is invertible and A−1

equivalently if the columns of A are an orthonormal basis of Rn. These matrices form a subgroup of the group of all invertible matrices, called the orthogonal group and denoted O(n).

Proposition 5.2. Let λ: Rn → Rn be a linear transformation with standard matrix A.

(i)λ is an isometry if and only if A is an orthogonal matrix.

(ii)The group of linear isometries of Rn is isomorphic to O(n).

Proof. If A is orthogonal, then for all v, w Rn,

ei žej = λ(ei )žλ(ej ) = Aei žAej = eTi (AT A)ej = the (i, j )-entry of A

106 5 SYMMETRY GROUPS IN THREE DIMENSIONS

for all i and j . It follows that AT A = I , so A is orthogonal, proving (i). But then the correspondence λ ↔ A between the linear transformation λ and its standard matrix A induces a group isomorphism between the (linear) isometries fixing the origin and the orthogonal matrices. This proves (ii).

Given a vector w Rn, define τw: Rn → Rn by τw(v) = v − w for all v Rn. Thus τw is the unique translation that carries w to 0. Because

ular, T (n) is an abelian group.

Theorem 5.3. For each n 1, T (n) is an abelian normal subgroup of E(n) and

α → the standard matrix of τα(0) Ž α

is a surjective group morphism E(n) → O(n) with kernel T (n).

Proof. Write G(n) for the group of all linear isometries of Rn. Observe that if α E(n), then τα(0) Ž α is linear (it is an isometry that fixes 0), so we have a map

φ: E(n) → G(n) given by φ(α) = τα(0) Ž α for all α E(n).

By Proposition 5.2 it suffices to show that φ is a surjective group morphism with kernel T (n). To see that φ is a group morphism, observe first that

unique translation that carries (α Ž β)(0) to 0. Hence φ is a group morphism.

If α E(n) and α(0) = w, the proof of Theorem 5.3 shows that τw Ž α G(n). Hence every isometry α of Rn is the composition of a linear isometry τw−1 Ž α followed by a translation τw.

Proposition 5.4. Every finite subgroup of isometries of n-dimensional space fixes at least one point.

Proof. Let G be a finite subgroup of isometries, and let x be any point of n-dimensional space. The orbit of x consists of a finite number of points that are permuted among themselves by any element of G. Since all the elements of G are rigid motions, the centroid of Orb x must always be sent to itself. Therefore, the centroid is a fixed point under G.

If the fixed point of any finite subgroup G of isometries is taken as the origin, then G is a subgroup of O(n), and all its elements can be written as orthogonal matrices. We now look at the structure of groups whose elements can be written as matrices.

MATRIX GROUPS

In physical sciences and in mathematical theory, we frequently encounter multiplicative group structures whose elements are n × n complex matrices. Such a group is called a matrix group if its identity element is the n × n identity matrix I . To investigate these groups, we have at our disposal, and shall freely apply, the machinery of linear algebra.

For example, if

then ({A0, A1, . . . , Am−1}, ·) is a real matrix group of order m isomorphic to Cm. The matrix Ak represents a counterclockwise rotation of the plane about the origin through an angle (2π k/m).

−

form a group under matrix multiplication. This is a complex matrix group of order 8 that is, in fact, isomorphic to the quaternion group Q of Exercise 3.47.

Since the identity of any matrix group is the identity matrix I and every element of a matrix group must have an inverse, every element must be a nonsingular matrix. All the nonsingular n × n matrices over a field F form a group (GL(n, F ), ·) called the general linear group of dimension n over F . Any matrix group over the field F must be a subgroup of GL(n, F ).

Proposition 5.5. The determinant of any element of a finite matrix group must be an integral root of unity.

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Proof. Let A be an element of a matrix group of order m. Then, by Corollary 4.10, Am = I . Hence (detA)m = detAm = detI = 1.

Hence, if G is a real matrix group, the determinant of any element of G is either +1 or −1. If G is a complex matrix group, the determinant of any element is of the form e2π ik/m.

The orthogonal group O(n) is a real matrix group, and therefore any element must have determinant +1 or −1. The determinant function

det: O(n) → {1, −1}

is a group morphism from (O(n), ·) to ({1, −1}, ·). The kernel, consisting of orthogonal matrices with determinant +1, is called the special orthogonal group of dimension n and is denoted by

SO(n) = {A O(n)|detA = +1}.

This is a normal subgroup of O(n) of index 2. The elements of SO(n) are called proper rotations, whereas the elements in the other coset of O(n) by SO(n), consisting of orthogonal matrices with determinant −1, are called improper rotations.

An n × n complex matrix A is called unitary if it is invertible and A−1 is the conjugate transpose of A. Thus the real unitary matrices are precisely the

The unitary group of dimension n, U(n), consists of all n × n complex unitary matrices under multiplication. The special unitary group, SU(n), is the subgroup of U(n) consisting of those matrices with determinant +1. The group SU(3) received some publicity in 1964 when the Brookhaven National Laboratory discovered the fundamental particle called the omega-minus baryon. The existence and properties of this particle had been predicted by a theory that used SU(3) as a symmetry group of elementary particles.

Proposition 5.6. If λ C is an eigenvalue of any unitary matrix, then |λ| = 1.

Proof. Let λ be an eigenvalue and x a corresponding nonzero eigenvector of

the unitary matrix A. Then Ax = λx, and since A preserves distances, ||x|| = ||Ax|| = ||λx|| = |λ|||x||. Since x is nonzero, it follows that |λ| = 1.

The group {Ak |k = 0, 1, . . . , m − 1} of rotations of the plane is a subgroup of SO(2), and the eigenvalues that occur are e±(2π ik/m). The matrix group isomorphic to the quaternion group Q is a subgroup of SU(2), and the eigenvalues that occur are ±1 and ±i.