Combinatorics Through Guided Discovery, 2004a

6.1. Permutation Groups 105
What you have observed about iota in Problem 251 is called the identity property
of iota. In the context of permutations, people usually call the function ι “the
identity” rather than calling it “iota.”
Since rotating first by 90 degrees and then by 270 degrees has the same effect
as doing nothing, we can think of the 270 degree rotation as undoing what the 90
degree rotation does. For this reason we say that in the rotations of the square, ρ 3 is
the “inverse” of ρ. In general, a function ϕ : T → S is called an inverse of a function
σ : S → T (the lower case Greek letter sigma) if ϕ ◦ σ = σ ◦ ϕ = ι. For a slower
introduction to inverses and practice with them, see Section A.1.3 in Appendix A.
Since a permutation is a bijection, it has a unique inverse, as in Section A.1.3. And
since the inverse of a bijection is a bijection (again, as in the Appendix), the inverse
of a permutation is a permutation.
We use ϕ −1 to denote the inverse of the permutation ϕ. We’ve seen that the
rotations of the square are functions that return the square to its original position
but may move the vertices to different places. In this way we create permutations
of the vertices of the square. We’ve observed three important properties of these
permutations.
• (Identity Property) These permutations include the identity permutation.
• (Inverse Property) Whenever these permutations include ϕ, they also include
ϕ −1 .
• (Closure Property) Whenever these permutations include ϕ and σ, they also
include ϕ ◦ σ.
A set of permutations with these three properties is called a permutation group2
or a group of permutations. We call the group of permutations corresponding to
rotations of the square the rotation group of the square. There is a similar rotation
group with n elements for any regular n-gon.
• Problem 252. If f : S → T, g : T → X, and h : X → Y, ish ◦ (g ◦ f )=
(h ◦ g) ◦ f ? What does this say about the status of the associative law
in a group of permutations?
ρ ◦ (σ ◦ ϕ) =(ρ ◦ σ) ◦ ϕ
• Problem 253.
(a) How should we define ϕ −n for an element ϕ of a permutation group? (h)
(b) Will the two standard rules for exponents
a m a n = a m+n and (a m ) n = a mn
2The concept of a permutation group is a special case of the concept of a group that one studies in
abstract algebra. When we refer to a group in what follows, if you know what groups are in the more
abstract sense, you may use the word in this way. If you do not know about groups in this more abstract
sense, then you may assume we mean permutation group when we say group.