Combinatorics Through Guided Discovery, 2004a

Chapter 6
Groups acting on sets
6.1 Permutation Groups
Until now we have thought of permutations mostly as ways of listing the elements
of a set. In this chapter we will find it very useful to think of permutations as
functions. This will help us in using permutations to solve enumeration problems
that cannot be solved by the quotient principle because they involve counting the
blocks of a partition in which the blocks don’t have the same size. We begin by
studying the kinds of permutations that arise in situations where we have used the
quotient principle in the past.
6.1.1 The rotations of a square
1
4
1
4
2
2
3
3
1
4
4
3
ρ
2
1
1
3
2 2
3 4
2
4
1
3
1
4
2
1
2
3
4
3
ρ 2
ρ 3 4
ρ
1
1
2
2
4 3
4 3
= identity
= ρ 0
Figure 6.1.1: The four possible results of rotating a square and maintaining its
position.
In Figure 6.1.1 we show a square with its four vertices labelled 1, 2, 3, and 4. We
have also labeled the spot in the plane where each of these vertices falls with the
same label. Then we have shown the effect of rotating the square clockwise through
90, 180, 270, and 360 degrees (which is the same as rotating through 0 degrees).
Underneath each of the rotated squares we have named the function that carries
out the rotation. We use ρ, the Greek letter pronounced “row,” to stand for a 90
degree clockwise rotation. We use ρ 2 to stand for two 90 degree rotations, and so
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