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Fearless Symmetry by Robert Gross, Avner Ash

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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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