FINITE GROUPS OF ORTHOGONAL OPERATORS ON THE PLANE

Theorem 6.4.1 Let G be a finite subgroup of the orthogonal group O2. There is an integer n such that G is one of the following groups:

(a) Cn': the cyclic group of order n generated by the rotation p#, where () = 2rr/n .

(b) Dn: the dihedral group of order ²n generated by two elements: the rotation po, where () = 2r r/n , and a reflection r' about a line .e through the origin.

We will take a moment to describe the dihedral group Dn before proving the theorem.

This group depends on the line of reflection, but if we choose coordinates so that .e becomes the horizontal axis, the group will contain our standard reflection r, the one whose matrix is

(6.4.2)

Then if we also write p for po, the 2n elements of the group will be the n powers p' of p and the n products p' r. The rule for commuting p and r is

where ^c = cos 0, ^s = sin 0, and ° = 21"(/n.

To conform with a more customary notation for groups, we denote the rotation P2 rr/n by x, and the reflection r by y.

Proposition 6.4.3 The dihedral group Dn has order 2n. It is generated by two elements x and y that satisfy the relations

Using the first two relations (6.4.3), the third one can be rewritten in various ways. It is equivalent to

For n > 3 , the dihedral and symmetric groups are not isomorphic, because Dn has order 2n, while Sn has order n!.

When n 2: 3, the elements of the dihedral group Dn are the orthogonal operators that carry a regular n-sided polygon A to itself - the group of symmetries of A. This is easy to see, and it follows from the theorem: A regular n-gon is carried to itself by the rotation by 2Jr/n about its center, and also by some reflections. Theorem 6.4.1 identifies the group of all symmetries as Dn.

The dihedral groups D i, Dz are too small to be symmetry groups of an n-gon in the usual sense. D i is the group {1, r} of two elements. So it is a cyclic group, as is Cz. But the element r of D i is a reflection, while the element different from the identity in Cz is the rotation with angle Jr. The group Dz contains the four elements {1, p, r, prJ, where p is the rotation with angle and p r is the reflection about the vertical axis. This group is isomorphic to the Klein four group.

If we like, we can think of Di and Dz as groups of symmetry of the 1-gon and 2-gon:

x n = ¹, ; = 1, y x = x 1 y.

The elements of Dn are

□

(6.4.4) x y x y = 1, and also to y x = x n 1y.

When n = 3, the relations are the same as for the symmetric group S3 (2.2.6).

Corollary 6.4.5 The dihedral group D 3 and the symmetric group S 3 are isomorphic. □

i-gon. 2-gon.

Section 6.4 Finite Groups of Orthogonal Operators on the Plane 165 We begin the proof of Theorem 6.4.1 now. A subgroup .r of the additive group of real numbers is called discrete if there is a (small) positive real number e such that every nonzero element c of .r has absolute value 2: e.

Lemma 6.4.6 Let .r be a discrete subgroup of lR.+ . Then either .r = {O}, or .r is the set Za of integer multiples of a positive real number a.

Proof. This is very similar to the proof of Theorem 2.3.3, that a nonzero subgroup of Z+ has the form Z n .

If a and b are distinct elements of .r, then since .r is a group, a — b is in .r, and

|a — b| > e. Distinct elements of .r are separated by a distance at least e. Since only finitely many elements separated by e can fit into any bounded interval, a bounded interval contains finitely many elements of .r.

Suppose that .r *{O}. Then .r contains a nonzero element b, and since it is a group, r contains -b as well. So it contains a positive element, say a'. We choose the smallest positive element a in .r. We can do this because we only need to choose the smallest element of the finite subset of .r in the interval 0 :: x :: a'.

We show that .r = Za. Since a is in .r and .r is a group, Z a C .r. Let b be an element of r . Then b = ra for some real number r. We take out the integer part of r, writing r = m + ro with m an integer and 0 :: ro < 1 . Since r is a group, b' = b - ma is in r and b' = roa. Then 0 :: b' < a. Since a is the smallest positive element in .r, b' must be zero. So b = m a, which is in Za. This shows that .r C Z a , and therefore that .r = Z a . □ Proof o f Theorem (6.4.1). Let G be a finite subgroup of O2. We want to show that G is C n or D n . We remember that the elements of O2 are the rotations pg and the reflections pgr.

Case 1: All elements of G are rotations.

We must prove that G is cyclic. Let .r be the set of real numbers ex. such that p a is in G. Then .r is a subgroup of the additive group lR.+, and it contains 2n. Since G is finite, .r is discrete. So .r has the form Zex.. Then G consists of the rotations through integer multiples of the angle ex.. Since 2 n is in .r, it is an integer multiple of ex.. Therefore ex. = 2 n /n for some integer n, and G = Cn.

Case 2: G contains a reflection.

We adjust our coordinates so that the standard reflection r is in G. Let H denote the subgroup consisting of the rotations that are elements of G . We apply what has been proved in Case 1 to conclude that H is the cyclic group generated by pg, for some angle () = 2 n /n . Then the 2n products p@ and p^r, for 0 : : k < n - 1, are in G , so G contains the dihedral group Dn. We claim that G = Dn, and to show this we take any element g of G. Then g is either a rotation or a reflection. If g is a rotation, then by definition of H , g is in H. The elements of H are also in Dn, so g is in Dn. If g is a reflection, we write it in the form p a r for some rotation p a . Since r is in G , so is the product g r = pa . Therefore p a is a power of

Po, and again, g is in Dn. □

Theorem 6.4.7 Fixed Point Theorem. Let G be a finite group of isometries of the plane.

There is a point in the plane that is fixed by every element of G, a point p such that g ( p ) = p for all g in G.

Proof This is a nice geometric argument. Let s be any point in the plane, and let S be the set of points that are the images of s under the various isometries in G. So each element s' of S has the form s' = g (s) for some g in G. This set is called the orbit of s for the action of G. The element s is in the orbit because the identity element 1 is in G, and s = l(s ). A typical orbit for the case that G is the group of symmetries of a regular pentagon is depicted below, together with the fixed point p of the operation.

Any element of G will permute the orbit S. In other words, if s' is in S and h is in G, then h (s') is in S: Say that s ' = g (s), with g in G. Since G is a group, h g is in G. Then h g (s) is in S and is equal to h (s').

*P

We list the elements of S arbitrarily, writing S = {si, . . . , s„}. The fixed point we are looking for is the centroid, or center ofgravity of the orbit, defined as

where the right side is computed by vector addition, using an arbitrary coordinate system in the plane.

Lemma 6.4.9 Isometries carry centroids to centroids: Let S = {s i, . . . , s„} be a finite set of points of the plane, and let p be its centroid, as defined by (6.4.8). Let m be an isometry. Let m (p ) = p ' and m (s,) = sf. Then p ' is the centroid of the set S' = {s^, . . . , s^}. □ The fact that the centroid of our set S is a fixed point follows. An element g of G permutes the orbit S. It sends S to S and therefore it sends p to p . □ Proof o f Lemma 6.4.9 This can be deduced by physical reasoning. It can be shown alge­

braically too. To do so, it suffices to look separately at the cases m = ta and m = cp, where cp is an orthogonal operator. Any isometry is obtained from such isometries by composition.

Case 1: m = ta is a translation. Then sf = s, + a and p ' = p + a. It is true that p ' = p + a = ± ( 0 i + a ) + • • . + (sn + a) ) = ±(s[ +---+ s'n).

Case 2: m = cp is a linear operator. Then

p' = cp(p) = CP(k(si + .. • + s„)) = k(cp(si) +---+ CP(Sn» = k ( s i + ---+ sn) . □

Section 6.5 Discrete Groups o f Isometries 167 By combining Theorems 6.4.1 and 6.4.7 one obtains a description of the symmetry groups of bounded figures in the plane.

Corollary 6.4.10 Let G be a finite subgroup of the group M of isometries of the plane.

If coordinates are chosen suitably, G becomes one of the groups Cn or Dn described in