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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