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Group theory for Maths, Physics and Chemistry students Arjeh Cohen Rosane Ushirobira Jan Draisma July 11, 2002
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Chapter 1
Introduction 1.1
Symmetry
Group theory is an abstraction of symmetry Symmetry is the notion that an object of study may look the same from different points of view. For instance, the chair in Figure 1.1 looks the same as its reflection in a mirror that would be placed in front of it, and our view on the wheel depicted next to the chair doesn’t change if we rotate our point of view over π/6 around the shaft. But rather than changing viewpoint ourselves, we think of an object’s symmetry as transformations of space that map the object ‘into itself’. What do we mean when we say that an object is symmetric? To answer this question, consider once more the chair in Figure 1.1. In this picture we see a plane V cutting the chair into two parts. Consider the transformation r of threedimensional space that maps each point p to the point p0 constructed as follows:
Figure 1.1: Bilateral and rotational symmetry. From: [7].
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CHAPTER 1. INTRODUCTION
let l be the line through p and perpendicular to V . Denoting the distance of a point x to V by d(x, V ), p0 is the unique point on l with d(p0 , V ) = d(p, V ) and p 6= p0 . The map r is called the reflection in V . Now apply r to all points in the space occupied by the chair. The result is a new set of points, but because of the choice of V it equals the old space occupied by the chair. For this reason, we call the chair invariant under r, or say that r is a symmetry (transformation) of the chair. Transforming any point in space to its reflection in the mirror W and next rotating it over an angle of π around the axis V ∩W , gives the same result as the reflection r in V . This is an illustration of what we observed about symmetry: the change of view on t