6 Permutation Groups

Home , Permutation group, Symmetric group

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Let S be a nonempty set and M(S) be the collection of all mappings from S into S. In this section, we will emphasize on the collection of all invertible mappings from S into S. The elements of this set will be called permutations because of Theorem 2.4 and the next deﬁnition.

Deﬁnition 6.1 Let S be a nonempty set. A one-to-one mapping from S onto S is called a permutation.

Consider the collection of all permutations on S. Then this set is a group with respect to composition.

Theorem 6.1 The set of all permutations of a nonempty set S is a group with respect to composition. This group is called the symmetric group on S and will be denoted by Sym(S).

Proof. By Theorem 2.4, the set of all permutations on S is just the set I(S) of all invertible mappings from S to S. According to Theorem 4.3, this set is a group with respect to composition.

Deﬁnition 6.2 A group of permutations , with composition as the operation, is called a permutation group on S.

Example 6.1 1. Sym(S) is a permutation group. 2. The collection L of all invertible linear functions from R to R is a permutation group with respect to composition.(See Example 4.4.) Note that L is

1 a proper subset of Sym(R) since we can ﬁnd a function in Sym(R) which is not in L, namely, the function f(x) = x3. This example shows that, in general, a permutation group on S needs not contain all the permutations on S. Example 6.2 Let S = {1, 2, 3}. There are six permutations on S. We will represent these permutations using the two-row form as follows: 1 2 3 1 2 3 1 2 3 ρ = , ρ = , ρ = 1 1 2 3 2 2 3 1 3 3 1 2 1 2 3 1 2 3 1 2 3 ρ = , ρ = , ρ = 4 2 1 3 5 3 2 1 6 1 3 2 In composing permutations we always follow the same convention we use in composing any other mappings: read from right to left. Thus, 1 2 3 1 2 3 1 2 3 ◦ = 3 1 2 3 2 1 2 1 3 That is, 1 → 3 → 2, 2 → 2 → 1, 3 → 1 → 3. Example 6.3 0 Let S = {1, 2, 3}. Then Sym(S) = {ρ1, ρ2, ρ3, ρ4, ρ5, ρ6}, where the ρ s are deﬁned in Example 6.2. Let’s construct the Cayley table for this group.

◦ ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ1 ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ2 ρ2 ρ3 ρ1 ρ5 ρ6 ρ4 ρ3 ρ3 ρ1 ρ2 ρ6 ρ4 ρ5 ρ4 ρ4 ρ6 ρ5 ρ1 ρ3 ρ2 ρ5 ρ5 ρ4 ρ6 ρ2 ρ1 ρ3 ρ6 ρ6 ρ5 ρ4 ρ3 ρ2 ρ1 Notice that the permutation 1 2 3 ρ = 1 1 2 3 is the identity mapping of Sym(S). Moreover,

1 2 3 −1 1 2 3 = 2 3 1 3 1 2

2 Thus, the inverse of an element is obtained by reading from the bottom entry to the top entry rather than from top to bottom:if 1 appears beneath 3 in ρ2 −1 then 3 appears beneath 1 in ρ2 . We will denote the above group by S3. In general, if S = {1, 2, ··· , n} then the symmetric group on S will be denoted by Sn.

The number of elements of Sn is found in the following theorem. Theorem 6.2 The order of Sn is n!, where 0! = 1! = 1 and n! = n(n − 1)(n − 2) ··· 2 · 1. Proof. The proof involves the following counting principle: If a decision consists of two steps, if the first step can be done in r different ways and the second step can be done in s different ways then the decision can be made in rs different ways. The problem of computing the number of elements of Sn is the same as the problem of computing the number of different ways the integers 1, 2, ··· , n can be placed in the n blanks indicated(with each number used only once) 1 2 3 ··· n ··· Filling the blanks from the left, we see that the first blank can be filled with n different ways. Once this is completed, the second blank can be filled in n − 1 ways, the third in n − 2 ways and so on. Thus, by the principle of counting, there are n(n − 1)(n − 2) ··· 2 · 1 = n! ways of filling the blanks. In conclusion, |Sn| = n!

Now, since S1 = {(1)} then S1 with respect to composition is commuta- tive. Similarly, since (1)(12) = (12)(1) then S2 = {(1), (12)} is also Abelian. Unfortunately, this is not true anymore for |S| > 2. Theorem 6.3 Sn is non-Abelian for n ≥ 3. Proof. All that we need to do here is to ﬁnd two permutations α and β in Sn with n ≥ 3 such that α ◦ β 6= β ◦ α. Indeed, consider the permutations 1 2 3 4 5 ··· n 1 2 3 4 5 ··· n α = and β = 1 3 2 4 5 ··· n 3 2 1 4 5 ··· n

3 Then 1 2 3 4 5 ··· n 1 2 3 4 5 ··· n α◦β = and β ◦α = 2 3 1 4 5 ··· n 3 1 2 4 5 ··· n so that α ◦ β 6= β ◦ α

Cycle Notation for Permutations The cycle notation for permutations can be thought as a condensed way to write permutations. Here is how it works. Let α ∈ Sn be the permutation

α(a1) = a2, α(a2) = a3, ··· , α(ak) = a1 and α(ai) = ai for i = k + 1, ··· , n, where a1, a2, ··· , an ∈ {1, 2, 3, ··· , n}. That is, α follows the circle pattern shown in Figure 6.1

Figure 6.1 Such a permutation is called a cycle of length k or simply a k-cycle. We will write

α = (a1a2a3 ··· ak) (1) Let us elaborate a little further on the notation employed in (1). The cycle notation is read from left to right, it says α takes a1 into a2, a2 into a3, etc., and finally ak, the last symbol, into a1, the first symbol. Moreover, α leaves all the other elements not appearing in the representation (1) fixed. Example 6.4 The permutation 1 2 3 4 5 6 7 α = 1 6 3 7 5 4 2

4 can be represented as a 4-cycle

α = (2647).

Note that one can write the same cycle in many ways using this type of notation. α = (2647) = (6472) = (4726) = (7264)

Remark 6.1 A k-cycle can be written in k diﬀerent ways, since

(a1a2 . . . ak) = (a2a3 . . . aka1) = ··· = (aka1 . . . ak−1).

Example 6.5 It is easy to write the inverse of a cycle. Since α(ak) = ak+1 implies −1 α (ak+1) = ak, we only need to reverse the order of the cyclic pattern. For example, (2647)−1 = (7462).

Example 6.6 Multiplication of cycles is performed by applying the right permutation ﬁrst. Consider the product in S5

(12)(245)(13)(125)

Reading from right to left

1 7−→ 2 7−→ 2 7−→ 4 7−→ 4 so 1 7−→ 4. Now 4 7−→ 4 7−→ 4 7−→ 5 7−→ 5 so 4 7−→ 5. Next 5 7−→ 1 7−→ 3 7−→ 3 7−→ 3

5 so 5 7−→ 3. Then 3 7−→ 3 7−→ 1 7−→ 1 7−→ 2 so 3 7−→ 2. Finally 2 7−→ 5 7−→ 5 7−→ 2 7−→ 1 so 2 7−→ 1. Since all the elements of A = {1, 2, 3, 4, 5} have been accounted for, we have (12)(245)(13)(125) = (14532).

Remark 6.2 A 1-cycle of Sn is the identity of Sn and is denoted by (1). Of course, (1) = (2) = (3) = ··· = (n).

Now not all permutations are cycles; for example, the permutation

1 2 3 4 5 6 7 8 α = 2 3 1 5 6 7 4 8 is not a cycle. However, one can check easily that

α = (123)(4567)

This suggests how we may extend the idea of cycles to cover all permutations.

Definition 6.3 If α and β are two cycles, they are called disjoint if the elements moved by one are left fixed by the other, i.e., their cycle representations contain different elements of the set S = {1, 2, 3 ··· , n}.

Example 6.7 The cycles (124) and (356) are disjoint whereas the cycles (124) and (346) are not since they have the number 4 in common.

Theorem 6.4 If α and β are disjoint cycles then αβ = βα.

6 Proof. Indeed, since the cycles α and β are disjoint, each element moved by α is ﬁxed by β and vice versa. Let α = (a1a2 ··· as) and β = (b1b2 ··· bt) where {a1, a2, ··· , as} ∩ {b1, b2, ··· , bt} = ∅.

(i) Let 1 ≤ k ≤ s. Then

(αβ)(ak) = α(β(ak)) = α(ak) = ak+1 and (βα)(ak) = β(α(ak)) = β(ak+1) = ak+1. (ii) Let 1 ≤ k ≤ t. Then

(αβ)(bk) = α(β(bk)) = α(bk+1) = bk+1 and (βα)(bk) = β(α(bk)) = β(bk) = bk+1.

(iii) Let 1 ≤ m ≤ n and m 6∈ {a1, a2, ··· , as, b1, b2, ··· , bt}. Then

(αβ)(m) = α(β(m)) = α(m) = m and (βα)(m) = β(α(m)) = β(m) = m. It follows from (i), (ii), and (iii) that αβ = βα.

Theorem 6.5 Every permutation of Sn is either a cycle or can be written uniquely, except for order of cycles or the diﬀerent ways a cycle is written, as a product of disjoint cycles. Proof. The proof is by induction on n. If n = 1 then there is only one permutation, and it is the cycle (1). Assume that the result is valid for all sets with fewer than n elements. We will prove that the result is valid for a set with n elements. Let σ ∈ Sn. If σ = (1) then we are done. Otherwise there exists a positive integer m such that σm−1(1) 6= 1 and σm(1) = 1. For example, if σ =

7 2 (145) ∈ S5 then m = 3 since σ(1) = 4, σ (1) = σ(σ(1)) = σ(4) = 5, and σ3(1) = σ(σ2(1)) = σ(5) = 1. Let Q = {1, σ(1), σ2(1), . . . , σm−1(1)}. If Q = {1, 2, . . . , n} then σ is the cycle σ = (1σ(1)σ2(1) . . . σm−1(1)). If Q 6= S, where S = {1, 2, ··· , n}, then

σ = (1σ(1)σ2(1) . . . σm−1(1))τ where τ is a permutation on the set S − Q = {t ∈ S : t 6∈ Q}. Since this set has order smaller than n, then by the induction hypothesis τ can be written as a product of disjoint cycles, say, τ = τ1τ2 . . . τk. Thus,

2 m−1 σ = (1σ(1)σ (1) . . . σ (1))τ1τ2 . . . τk

But this says that σ is expressed as a product of disjoint cycles. This com- pletes the induction step, and establishes the result for all n

Example 6.8 Let S = {1, 2, ..., 8} and let

1 2 3 4 5 6 7 8 α = 2 4 6 5 1 7 3 8

Start the first cycle with 1 and continue until we get back to 1, and then close the first cycle. Then start the second cycle with the smallest number not in the first cycle, continue until we get back to that number, and then close the second cycle, and so on to obtain

α = (1245)(367)(8)

It is customary to omit such cycles as (8), i.e., elements left ﬁxed by α, and write α simply as α = (1245)(367)