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§26 Permutations x26 Permutations Tom Lewis Fall Term 2010 Tom Lewis () x26 Permutations Fall Term 2010 1 / 16 Outline 1 Permutations 2 The symmetric group 3 Cycles 4 Transpositions 5 Inversions Tom Lewis () x26 Permutations Fall Term 2010 2 / 16 Permutations Definition (Permutation) Let A be a set. A permutation on A is a bijection from A to A. A convention! Permutations are usually denoted by lower-case Greek letters, for example, π, τ, σ, and θ. The identity permutation Following our text, we will denote the identity permutation by {, a dotless i. Tom Lewis () x26 Permutations Fall Term 2010 3 / 16 Permutations Problem Let A = f1; 2; 3; 4; 5; 6; 7g. Show that π = f(1; 3); (2; 1); (3; 4); (4; 2); (5; 6); (6; 5); (7; 7)g is a permutation. Note that we will often write π as 1 2 3 4 5 6 7 π = 3 1 4 2 6 5 7 Tom Lewis () x26 Permutations Fall Term 2010 4 / 16 Permutations Problem Let 1 2 3 4 5 6 7 1 2 3 4 5 6 7 π = and τ = 3 1 4 2 6 5 7 2 1 7 6 3 5 4 Find the permutations π ◦ τ, τ ◦ π, π−1, and τ −1. Tom Lewis () x26 Permutations Fall Term 2010 5 / 16 The symmetric group Definition (The symmetric group) The set of all permutations on f1; 2; 3;:::; ng is called the symmetric group and is denoted by Sn. Tom Lewis () x26 Permutations Fall Term 2010 6 / 16 The symmetric group What is a group? A group is a set with a binary operation (usually called multiplication). The set and the binary operation must satisfy the four group properties: closure, associativity, identity, and inverse. Here is what this means for Sn: Closure If π and τ are two elements in Sn, then the product π ◦ τ is also in Sn. Associativity For all π; τ; σ 2 Sn, (π ◦ τ) ◦ σ = π ◦ (τ ◦ σ): Identity If we let id denote the identity function on f1; 2; 3;:::; ng, then id ◦π = π ◦ id = π for every element π in Sn. −1 Inverse Each element of Sn has an inverse. If π 2 Sn, then π 2 Sn and π ◦ π−1 = π−1 ◦ π = id : Tom Lewis () x26 Permutations Fall Term 2010 7 / 16 Cycles Problem Let 1 2 3 4 5 6 7 π = 3 1 4 2 6 5 7 1 Show that the function values 1; π(1); π(2)(1); π(3)(1);::: are cyclical. 2 Express π as a collection of pairwise disjoint cycles. 3 Can we write any permutation as a collection of pairwise disjoint cycles? Tom Lewis () x26 Permutations Fall Term 2010 8 / 16 Cycles Theorem Let π be a permutation on A = f1; 2;:::; ng. Show the following: (k ) 1 For each x 2 A, there exists a smallest k0 ≥ 1 such that π 0 (x) = x. (k) 2 The relation i R j provided that there exists k 2 N such that π i = j is an equivalence relation on A. Theorem Every permutation of a finite set can be expressed as a collection of pairwise disjoint cycles. Furthermore, this representation is unique up to rearranging the cycles and the cyclic order of the elements within the cycles. Tom Lewis () x26 Permutations Fall Term 2010 9 / 16 Cycles Problem Let π; τ 2 S6 be given by π = (1; 2; 3)(4; 5; 6) and τ = (1; 6)(2; 3)(4; 5) Calculate π ◦ τ, τ ◦ π, π−1, τ −1. Represent each answer in cycle notation. Tom Lewis () x26 Permutations Fall Term 2010 10 / 16 Transpositions Example The permutation π = (1)(2)(3)(4; 8)(5)(6)(7) is an example of a transposition. Notice that all but two of the elements are fixed by π. Abuse of notation! Due to what can only be categorized as laziness, we will often compress π = (1)(2)(3)(4; 8)(5)(6)(7) to (4; 8). Tom Lewis () x26 Permutations Fall Term 2010 11 / 16 Transpositions Definition A transposition of Sn is a permutation for which all but two elements of the domain are fixed by π. Thus if π is a transposition of Sn, then there exist two elements i and j of f1; 2;:::; ng such that 1 π(i) = j and π(j) = i. 2 π(x) = x for all x not equal to i and j. Tom Lewis () x26 Permutations Fall Term 2010 12 / 16 Transpositions Problem Express π = (1; 3; 4; 2) as a composition of transpositions in two different ways. (Try to make the number of transpositions required of different lengths.) Theorem Let π be a permutation of a finite set. Then π can be expressed as a composition of transpositions. (Work by induction.) You've been warned This theorem is often expressed in the following language: Every permutation can be written as a product of transpositions. Tom Lewis () x26 Permutations Fall Term 2010 13 / 16 Transpositions Definition Let π be decomposed into a product of transpositions τn ◦ τn−1 ◦ · · · τ2 ◦ τ1. If the number of transpositions is an even number, then we say that the decomposition is even. If the number of transpositions is odd, then we say that the decomposition is odd. Lemma Every decomposition of the identity permutation is even. Note: We will delay the proof of this crucial result, but we will shamelessly use it to prove::: Theorem Let π 2 Sn. Any two decompositions of π must be either both even or both odd. In other words, the parity of a permutation is an invariant. Tom Lewis () x26 Permutations Fall Term 2010 14 / 16 Inversions Definition (Inversion) Let π 2 Sn and let i; j 2 f1; 2;:::; ng with i < j. The pair i; j is called an inversion in π if π(i) > π(j). Problem Count the number of inversions in 1 2 3 4 5 σ = 4 5 2 1 3 and 1 2 3 4 5 6 7 8 τ = 1 2 7 4 5 6 3 8 Tom Lewis () x26 Permutations Fall Term 2010 15 / 16 Inversions Theorem A transposition has an odd number of inversions. Lemma Every decomposition of the identity permutation is even. Tom Lewis () x26 Permutations Fall Term 2010 16 / 16.
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