Mathematical sets can be interesting and easy to deﬁne,

but very hard to understand in detail. For example, t he

set of all prime numbers is easy to deﬁne but is the

source of many unsolved problems in number theory.

If a set can be endowed with extra structure, it can

help our understanding. One very common kind of extra

structure is a “composition law” that turns the set into

a group. The deﬁnition of a group is the most common

way mathematicians have of formalizing the concept of

symmetry.

The concept of a group is necessary for the representa-

tion theory we are developing. All of our objects are groups

of one kind or another. There is a lot known about groups

in general and about special kinds of groups in particular,

and we will be able to deploy all this knowledge when we

can impose a group structure on the objects we want to

learn about.

In this chapter, we state the deﬁnition of group and

look in detail at a particular example, the group SO(3) of

rotations of a s phere.

1

We chose this example because it is

easy to visualize but complicated enough to give you the

full ﬂavor of “groupness.”

1

SO(3) stands for “special orthogonal group in three dimensions.”

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