Before diving back into number theory, we explain the

key concept that puts together groups, permutations,

and matrices: group representations. We will give several

different examples. Except for the last example, which

uses elliptic curves, we mostly keep away from number

theory in this chapter, so the unadorned concept of group

representation can stand out more clearly.

Our central example is that of the symmetries of a

tetrahedron, which shows how geometrical symmetry can

be interpreted using group theory, and then be understood

further using representation theory. This example is a

simple version of many other, more complicated, geomet-

rical symmetry groups and their representations, which

could be the subject matter of a different book.

Morphisms of Groups

At last, we come to the heart of this book, or at least the peri-

cardium. A group representation is nothing more nor less than a

morphism from one group to another group. The reason we use the

special term “representation” is that the target group is chosen to

be one we are especially comfortable with, or one whose properties

are important for understanding the source group. The two types

of representations we will look at are permutation representations

and linear representations.

136 CHAPTER 12

What is a morphism from the group H to the group K?Itis

a function, call it f (x), for example, from H to K with just one

property:

f (x ◦ y) = f (x) ◦ f (y) (12.1)

for any two elements x and y in H.Inotherwords:f is a rule that

assigns to any element, say x,fromH, exactly one element f (x)from

K. And it does this in such a wa y that (12.1) holds. Note that in

(12.1), the little circle in x ◦ y refers to the group law in H, while

that in f (x) ◦ f (y) refers to the group law in K. W e summarize all of

this as follows: f : H → K is a morphism of groups.

EXAMPLE: We start with an example that captures three of

the most basic facts about multiplication of numbers:

•

positive × positive = positive.

•

positive × negative = negative.

•

negative × negative = positive.

In this example, our source group H is R

×

, the set that

contains all real numbers other than 0, with the group law of

multiplication. The target group K is {+1, −1}, again with the

group law of multiplication. The morphism f is the following

rule:

f (x) =

+1ifx > 0.

−1ifx < 0.

You can check that the equation f (xy) = f (x)f (y) will always

be true, if x and y are any nonzero real numbers.

In this equation, we throw away a lot of information about

the source group R

×

and preserve only the sign of the

number. Our representation aggregates the source group into

two clumps—positive numbers and negative numbers—and

ignores all of the other information about real numbers. It

captures the rules for multiplication of signed numbers at the

expense of forgetting about all of the other properties of real

numbers.

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