Combinatorics Through Guided Discovery, 2004a

104 6. Groups acting on sets
on. We can think of the function ρ as a function on the four element set1 {1, 2, 3, 4}.
In particular, for any function ϕ (the Greek letter phi, usually pronounced “fee,”
but sometimes “fie”) from the plane back to itself that may move the square around
but otherwise leaves it in the same place, we let ϕ(i) be the label of the place where
vertex previously in position i is now. Thus ρ(1) = 2, ρ(2) = 3, ρ(3) = 4 and
ρ(4) = 1. Notice that ρ is a permutation on the set {1, 2, 3, 4}.
• Problem 248. The composition f ◦ g of two functions f and g is defined by
f ◦ g(x) = f (g(x)). Is ρ 3 the composition of ρ and ρ 2 ? Does the answer
depend on the order in which we write ρ and ρ 2 ?Howisρ 2 related to ρ?
• Problem 249. Is the composition of two permutations always a permutation?
In Problem 248 you see that we can think of ρ 2 ◦ ρ as the result of first rotating
by 90 degrees and then by another 180 degrees. In other words, the composition
of two rotations is the same thing as first doing one and then doing the other. Of
course there is nothing special about 90 degrees and 180 degrees. As long as we
first do one rotation through a multiple of 90 degrees and then another rotation
through a multiple of 90 degrees, the composition of these rotations is a rotation
through a multiple of 90 degrees.
If we first rotate by 90 degrees and then by 270 degrees then we have rotated
by 360 degrees, which does nothing visible to the square. Thus we say that ρ 4 is
the “identity function.” In general the identity function on a set S, denoted by ι
(the Greek letter iota, pronounced eye-oh-ta) is the function that takes each element
of the set to itself. In symbols, ι(x) =x for every x in S. Of course the identity
function on a set is a permutation of that set.
6.1.2 Groups of Permutations
• Problem 250. For any function ϕ from a set S to itself, we define ϕ n (for
nonnegative integers n) inductively by ϕ 0 = ι and ϕ n = ϕ n−1 ◦ ϕ for every
positive integer n. Ifϕ is a permutation, is ϕ n a permutation? Based on your
experience with previous inductive proofs, what do you expect ϕ n ◦ ϕ m to
be? What do you expect (ϕ m ) n to be? There is no need to prove these last
two answers are correct, for you have, in effect, already done so in Chapter 2.
• Problem 251. If we perform the composition ι ◦ ϕ for any function ϕ from
S to S, what function do we get? What if we perform the composition ϕ ◦ ι?
1What we are doing is restricting the rotation ρ to the set {1, 2, 3, 4}.