FINITE ROTATION GROUPS IN THREE DIMENSIONS

We take Cn to be the rotation group of a regular n-agonal cone whose base is a regular n-gon. (The sloping edges of the cone must not be equal to the base edges if n = 3.) Dn is the rotation group of a regular n-agonal cylinder whose base is a regular n-gon. (The vertical edges must not be equal to the base edges if n = 4.)

Each stabilizer group, Stab x, is a cyclic subgroup of rotations of the solid about the axis through x. The orbit of x, Orb x, is the set of poles of the same type as x. As a check on the number of elements in the stabilizers and orbits, we have |G| = |Stab x||Orb x| for each pole x.

For example, the cube has three types of poles and four types of nontrivial elements in its rotation group; these are illustrated in Figure 5.7.

Theorem 5.15. Any finite subgroup of SO(3) is isomorphic to one of

Cn(n 1), Dn(n 2), A4, S4 or A5.

Proof. Let G be a finite subgroup of SO(3). Choose a set of poles x1, . . . , xr , one from each orbit. Let pi = |Stab xi | and qi = |Orb xi |, so that pi qi = n = |G|.

Each nonidentity element of G has two poles; thus the total number of poles, counting repetitions, is 2(n − 1). The pole xi occurs as a pole of a nonidentity element pi − 1 times. There are qi poles of the same type as xi . Therefore, the total number of poles, counting repetitions, is

2(n − 1) = qi (pi − 1) = (n − qi ),

118 5 SYMMETRY GROUPS IN THREE DIMENSIONS

If G is not the trivial group, n 2 and 1 2 − 2 < 2. Since xi is a pole of n

some nonidentity element, Stab xi contains a nonidentity element, and pi 2.

that x1 = x2, and there is just one axis of rotation. Therefore, G is isomorphic to the cyclic group Cn.

np1 p2 p3

Case (ii). p1 = 2, p2 = 3, p3 = 3, n = 12, q1 = 6, q2 = 4, and q3 = 4. Case (iii). p1 = 2, p2 = 3, p3 = 4, n = 24, q1 = 12, q2 = 8, and q3 = 6. Case (iv). p1 = 2, p2 = 3, p3 = 5, n = 60, q1 = 30, q2 = 20, and q3 = 12.

2

> 0.

n

of O(3) is isomorphic to one of the groups in Theorem 5.15 or contains one of these groups as a normal subgroup of index 2. See Coxeter [27, Sec. 15.5] for a more complete description of these improper rotation groups.

CRYSTALLOGRAPHIC GROUPS

This classification of finite symmetries in R3 has important applications in crystallography. Many chemical substances form crystals and their structures take the forms of crystalline lattices. A crystal lattice is always finite, but in order to study its symmetries, we create a mathematical model by extending this crystal lattice to infinity. We define an ideal crystalline lattice to be an infinite set of points in R3 of the form

n1a1 + n2a2 + n3a3,

where a1, a2, and a3 form a basis of R3 and n1, n2, n3 Z. Common salt forms a cubic crystalline lattice in which a1, a2, and a3 are orthogonal vectors of the same length. Figure 5.10 illustrates a crystalline lattice.

This use of the term lattice is not the same as that in Chapter 2, where a lattice referred to a special kind of partially ordered set. To avoid confusion, we always use the term crystalline lattice here.

A subgroup of O(3) that leaves a crystalline lattice invariant is called a crystallographic point group. This is a finite subgroup of O(3) because there are only a finite number of crystalline lattice points that can be the images of a1, a2, and a3 when the origin is fixed.

However, not all finite subgroups of O(3) are crystallographic point groups. Suppose that A SO (3) leaves a crystalline lattice L invariant. Then, by Theorem 5.10, A is a rotation through an angle θ , and the trace of A is 1 + 2 cos θ . If we choose a basis for R3 consisting of the vectors a1, a2, a3 of the crystalline lattice L, the matrix representing A will have integer entries. The trace is invariant under change of basis, so the trace of A must be an integer. Hence 2 cos θ

Figure 5.10. Crystalline lattice.

must be integral, and θ must be either kπ/2 or kπ/3, where k Z. It follows that every element of a crystallographic point group in SO(3) can only contain elements of order 1, 2, 3, 4, or 6.

It can be shown that every crystallographic point group in SO(3) is isomorphic to one of C1, C2, C3, C4, C6, D2, D3, D4, D6, A4, or S4.

If we allow reflections, the only other such groups in O(3) must contain one of these groups as a normal subgroup of index 2. Every one of these groups occurs in nature as the point group of at least one chemical crystal. See Coxeter [27, Sec. 15.6] or Lomont [31, Chap. 4, Sec. 4].

EXERCISES

Find the group of proper rotations and the group of all rotations of the figures in Exercises 5.1 to 5.7.

5.5. 5.6. 5.7.

5.8.Let G be the subgroup of O(2) isomorphic to Dn. Find two matrices A and B so that any element of G can be written as a product of A’s and B’s.

5.9.What is the group of proper rotations of a rectangular box of length 3 cm, depth 2 cm, and height 2 cm?

Find the proper rotation group of the 13 Archimedean solids in Exercises 5.10 to 5.22. All the faces of these solids are regular polygons and all the vertices are similar. (See Cundy and Rollet [28] for methods on how to construct these solids.)

6

´

POLYA–BURNSIDE METHOD OF ENUMERATION

This chapter provides an introduction to the Polya´–Burnside method of counting the number of orbits of a set under the action of a symmetry group. If a group G acts on a set X and we know the number of elements of X, this method will enable us to count the number of different types of elements of X under the action of G.

For example, how many different chemical compounds can be obtained by attaching a CH3 or H radical to each carbon atom in the benzene ring of Figure 6.3? There are 26 different ways of attaching a CH3 or H radical on paper, but these do not all give rise to different compounds because many are equivalent under a symmetry. There are six different ways of attaching one CH3 radical and five H radicals, but they all give rise to the same compound. The dihedral group D6 acts on the 26 ways of attaching the radicals, and the number of different compounds is the number of orbits under the action of D6, that is, the number of formulas that cannot be obtained from each other by any rotation or reflection.

We have seen that the number of different switching circuits that can be obtained with n switches is 22n . This number grows very quickly as n becomes large. Table 2.11 gives the 16 switching functions of two variables; when n = 3, there are 256 different circuits, and when n = 4, there are 65,536 different circuits. However, many of these circuits are equivalent if we change the labels of the switches. That is, the symmetric group, Sn, acts on the 22n different circuits by permuting the labels of the switches. The number of nonequivalent circuits is the number of orbits under the action of Sn.

BURNSIDE’S THEOREM

Let G be a finite group that acts on a finite set X. The following theorem describes the number of orbits in terms of the number of elements left fixed

Modern Algebra with Applications, Second Edition, by William J. Gilbert and W. Keith Nicholson ISBN 0-471-41451-4 Copyright 2004 John Wiley & Sons, Inc.