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No credit card required 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
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
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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