GROUPS 19

of physics were made through idealized approximations , so that

patterns could be observed, for example, the inclined plane of

Galileo, or the thought experiments of Einstein.

Group theory gives us a mathematical way to deal with sym-

metry. But the actual earth has no exact symmetries: No rotation

(except the neutral, do-nothing rotation) takes it into a position

where it exactly coincides with itself. The idealized sphere, on the

other hand, is very symmetrical. This is reﬂected in the fact that

SO(3) has many different elements—all rotations, which smoothly

turn the sphere within its container.

Digression: Lie Groups

Our i dealized earth turns smoothly all the time at exactly the same

rate. There should be a concept of “inﬁnitesimal rotation”—how

much the earth turns in just an instant of time. This would be an

“inﬁnitesimal” element in SO(3), something like an angular velocity

around a certain axis. Any actual rotation should be buildable by

integrating some inﬁnitesimal rotation. This is the germ of the

theory of Lie groups and Lie algebras, named for the Norwegian

mathematician Sophus Lie (pronounced “lee”). Lie group theory is

thus a kind of marriage between calculus and group theory. It has

a place in number theory too, but that is beyond the scope of our

book.

A group such as SO(3) that has inﬁnitesimal generators is called

a continuous group. The other main type of group is a called a

discrete group.Inthelatter,thereisnosmoothpathfromone

element of the group to another. An example of a discrete group

is the set of integers, where we “compose” two integers by adding

them together. The neutral element is 0 and the inverse of x is −x.

The next type of group we will study, a permutation group, is also a

discrete group.

Other examples of Lie groups include the set of rigid motions of

space to itself, used in crystallography and Newtonian physics; the

set of Lorentz transformations of ﬂat spacetime, used in relativistic

physics; the set of rotations of a circle, also called SO(2); and the

20 CHAPTER 2

set of real numbers with composition given by ordinary addition.

(Composition in the ﬁrst three of these examples is given by doing

ﬁrst one motion and then the second, similar to composition in

SO(3).)

Lie groups turn up when we study a geometric object with a lot

of symmetry, such as a sphere, a circle, or ﬂat spacetime. Because

there is so much symmetry, there are many functions from the

object to itself that preserve the geometry, and these functions

become the elements of the group. As we will see, discrete groups

can be also used to keep track of symmetries.

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