GROUP REPRESENTATIONS 139

DEFINITION: A linear representation (or matrix

representation)ofagroupH is a morphism from H to a

matrix group GL(n, R), where R is some number system.

For instance, suppose that G is the absolute Galois group of Q.

Look back at our discussion in chapter 8 about how any element g in

G permutes the roots of any Z-polynomial. So if we pick a particular

polynomial, we can ask how any element of G permutes its roots. We

will explain this example in detail in chapter 14. You can also look

at our description of the n-torsion of an elliptic curve in chapter 9.

There, too, we can think of an element of G as permuting the

n-torsion. In this case, we can also produce linear representations

of G. See the end of this chapter and chapter 18 for more details.

A

4

, Symmetries of a Tetrahedron

This is a very long example. If the going gets a bit dense, don’t

worry; we will not need the details of this example in any other

part of this book. Our source group in this example is a permutation

group called A

4

. First, we will tell you about the group abstractly,

and then we will give you some other wa ys to think about it by

using representation theory.

Our group A

4

is a group of permutations of the set {1, 2, 3, 4},but

it does not contain every permutation (or else it would be

{1,2,3,4}

).

In fact, A

4

contains exactly 12 permutations, or half of all of the

permutations in

{1,2,3,4}

.

To start, A

4

contains every permutation that leaves one number

ﬁxed, and cyclically permutes the other three numbers. For exam-

ple, A

4

contains the permutation that sends 2 to 2, and cyclically

permutes 1, 3, and 4, meaning that 1 → 3 → 4 → 1:

1 → 3

2 → 2

3 → 4

4 → 1.

140 CHAPTER 12

The short notation for writing this permutation is (134), where

the omission of the number 2 means that 2 is not changed by the

permutation.

There are eight such permutations.

EXERCISE: List all eight of the permutations that ﬁx one

element of {1, 2, 3, 4} and cyclically permute the other three

numbers.

SOLUTION: It would consume too much space to list these

permutations using the arrow notation. Using the shorter

cycle notation, the eight permutations are (123), (132), (134),

(143), (124), (142), (234), and (243).

Remember that a group must also contain the neutral element

e (which leaves every element of {1, 2, 3, 4} ﬁxed), so you now know

nine elements of A

4

. The other three elements can be found because

our group must be closed under the composition law.

The result of (123) ◦ (124) (which, you remember, means ﬁrst

doing the permutation (124) and then the permutation (123)) is:

1 → 3

2 → 4

3 → 1

4 → 2.

The way to write this in cycle notation is (13)(24).

There are two other permutations that result from combining

the elements of our group: (12)(34) and (14)(23). Here is the list of

all of the permutations in A

4

: e, (123), (132), (134), (143), (124), (142),

(234), (243), (12)(34), (13)(24), (14)(23).

EXERCISE: Find a permutation of {1, 2, 3, 4} that is not in A

4

.

SOLUTION: Here are all of the permutations in

{1,2,3,4}

that

are not in A

4

: (12), (13), (14), (23), (24), (34), (1234), (1324),

(1423), (1243), (1342), and (1432).

GROUP REPRESENTATIONS 141

Figure 12.1: A regular tetrahedron

This is pretty abstract stuff. Fortunately, there is another way to

think about A

4

.

DEFINITION: A regular tetrahedron is a solid ﬁgure with

four congruent equilateral triangles as its faces.

You can see a picture of a tetrahedron in F igure 12.1.

Number the vertices of the tetrahedron from 1 to 4. For the sake

of illustration, label the top vertex with the number 1, and the three

on the base 2, 3, and 4 (counterclockwise).

Now imagine rotating the tetrahedron so that when you are

done, the tetrahedron still looks the same, except that the vertices

may have moved around. For example, you may keep the vertex

numbered 1 ﬁxed, and spin the tetrahedron 120

◦

counterclockwise,

which will shift the other three vertices. The effect of this spinning

on the vertices is the permutation

1 → 1

2 → 3

3 → 4

4 → 2.

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