What is a group?

A group G is a set of objects f, g, h, … and an operation called multiplication such that:

1. if f ∈ G and g ∈ G, the product fg ∈ G (closure);

2. if f, g, and h are in G, then f(gh) = (fg)h (associativity);

3. there is an identity e ∈ G such that if g ∈ G, then ge = eg = g;

4. every g ∈ G has an inverse g −1 ∈ G such that gg −1 = g −1 g = e.

Physical transformations naturally form groups. The product T′ T represents the transformation T followed by the transformation T′. And both T″ (T′ T) and (T″ T′) T represent the transformation T followed by the transformation T′ and then by T″. So transformations are associative. The identity element e is the null transformation, the one that does nothing. The inverse T −1 is the transformation that reverses the effect of T. Such a set {T} of transformations will form a group if any two successive transformations is a transformation in the set (closure). Closure occurs naturally when the criterion for membership in the group is that a transformation not change something. For if both T and T′ leave that thing unchanged, then so will their product T′ T.

Example 10.1 (Groups of coordinate transfor mations) The set of all transformations that leave invariant the distance from the origin of every point in n-dimensional space is the group O(n) of rotations and reflections. The rotations in R n form the group SO(n).

To the students: you will find some physics crammed in amongst the mathematics. Don't let the physics bother you. As you study the math, you'll learn some physics without extra effort. The physics is a freebie. I have tried to explain the math you need for physics and have left out the rest.

To the professors: the book is for students who also are taking mechanics, electrodynamics, quantum mechanics, and statistical mechanics nearly simultaneously and who soon may use probability or path integrals in their research. Linear algebra and Fourier analysis are the keys to physics, so the book starts with them, but you may prefer to skip the algebra or postpone the Fourier analysis. The book is intended to support a one- or two-semester course for graduate students or advanced undergraduates. The first seven, eight, or nine chapters fit in one semester, the others in a second. A list of errata is maintained at panda.unm.edu/cahill, and solutions to all the exercises are available for instructors at www.cambridge.org/cahill.

Several friends – Susan Atlas, Bernard Becker, Steven Boyd, Robert Burckel, Sean Cahill, Colston Chandler, Vageli Coutsias, David Dunlap, Daniel Finley, Franco Giuliani, Roy Glauber, Pablo Gondolo, Igor Gorelov, Jiaxing Hong, Fang Huang, Dinesh Loomba, Yin Luo, Lei Ma, Michael Malik, Kent Morrison, Sudhakar Prasad, Randy Reeder, Dmitri Sergatskov, and David Waxman – have given me valuable advice.

Points and coordinates

A point on a curved surface or in a curved space also is a point in a higher-dimensional flat space called an embedding space. For instance, a point on a sphere also is a point in three-dimensional euclidean space and in fourdimensional space-time. One always can add extra dimensions, but it's simpler to use as few as possible, three in the case of a sphere.

On a sufficiently small scale, any reasonably smooth space locally looks like n-dimensional euclidean space. Such a space is called a manifold. Incidentally, according to Whitney's embedding theorem, every n-dimensional connected, smooth manifold can be embedded in 2n-dimensional euclidean space ℝ2n. So the embedding space for such spaces in general relativity has no more than eight dimensions.

We use coordinates to label points. For example, we can choose a polar axis and a meridian and label a point on the sphere by its polar and azimuthal angles (θ, ϕ) with respect to that axis and meridian. If we use a different axis and meridian, then the coordinates (θ′, ϕ′) for the same point will change. Points are physical, coordinates are metaphysical. When we change our system of coordinates, the points don't change, but their coordinates do.

Most points p have unique coordinates xi (p) and x′i (p) in their coordinate systems. For instance, polar coordinates (θ, ϕ) are unique for all points on a sphere – except the north and south poles which are labeled by θ = 0 and θ = π and all 0 ≤ ϕ < 2π.