In this chapter we deﬁne and discuss permutation groups.

In addition to being useful examples of groups, they are

essential for our later deﬁnition of Galois groups. Galois

groups are permutation groups of a certain kind: They

permute roots of polynomials.

The abc of Permutations

Our next type of group goes back to the idea of one-to-one correspon-

dence.

1

We start with a ﬁnite set, for example, {a, b, c}.Onething

they do in elementary school is to ﬁgure out all the possible ways

of ordering this set. There are six ways: a, b, c; a, c, b; b, a, c; b, c, a;

c, a, b; c, b, a. We can view any one of these orderings as the result

of a one-to-one correspondence of this set with itself. For example,

the third ordering can be viewed as the result of

a → b

b → a

c → c.

1

Remember that a one-to-one correspondence (deﬁned in chapter 1) is a special kind

of function. Perhaps we should call it a “one-to-one transformation,” but that means

something else in standard mathematical jargon, so we will stick with the traditional

and slightly misleading terminology.

22 CHAPTER 3

We obtained this result by lining up the letters a, b, c in the original

order (vertically for ease of viewing) and lining them up in the new

order in the next column.

We think of a one-to-one correspondence in a dynamic way,

following the model of our group SO(3). The elements of that

group were deﬁned in terms of actions that we took—rotating

the sphere—and taking note only of the end result of the action.

Similarly, we consider a permutation of the set {a, b, c} to be some

action we perform on the letters a, b,andc, and the end result will

be the letters a, b,andc in a new order.

For example, we deﬁne an action called “g”: g sends a to b, b to a,

and c to c. There are several ways to think of the action. One is

as a box, with an input slot on top, and an output slot on the

bottom. We call this the “g-box,” because it performs the action of

the permutation g.Ifwedropa into the slot, b comes out. If we drop

b in, a comes out. If we drop c in, c comes out.

Note that the g-box represents a function. It is not itself a

permutation; it is a box. But it tells us how to get a permutation:

Drop in a, b,andc successively, and out comes b, a,andc

successively. We say that the permutation is deﬁned by this box.

(If we just sit there dropping in a’s all day long, we will not ﬁnd

out what permutation is deﬁned by the box, but if we drop in a, b,

and

c successively, we will.) This box picture gives us a vivid way to

explain how to combine two permutations to get a third, as follows.

Each of the six possible orders of the three letters a, b, c deﬁnes

one of these boxes. So we have six boxes, and they correspond to the

six elements of this set of permutations. We combine elements of

this set via a composition law. For example, let g be the permutation

deﬁned above, and let h be the permutation deﬁned by the order c,

b, a, or, equivalently, by the one-to-one correspondence

a → c

b → b

c → a.

Then h ◦ g is deﬁned as follows: Take the g-box and put the h-box

underneath. Put an a into the g-box: out will drop b,andtheb will

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