Abstract Algebra, Lecture 5 Jan Snellman Abstract Algebra, Lecture 5 The Symmetric Permutations group Permutations 1 Groups of Jan Snellman Symmetries 1 Cayley's theorem Matematiska Institutionen | every group is a Link¨opingsUniversitet permutation group Link¨oping,spring 2019 Lecture notes availabe at course homepage http://courses.mai.liu.se/GU/TATA55/ Abstract Algebra, Lecture 5 Jan Snellman Summary The Symmetric group Permutations 1 The Symmetric group S2, S3, S4 Groups of Definition Even and Odd Permutations Symmetries Conjugation Cayley's theorem 3 Groups of Symmetries 2 Permutations | every group is a Linear Isometries permutation group S n The Dihedral groups Representations and notations Symmetry Groups of the Permutation Statistics Platonic Solids A note on left vs right Transpositions, k-cycles, 4 Cayley's theorem | every generating sets group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman Summary The Symmetric group Permutations 1 The Symmetric group S2, S3, S4 Groups of Definition Even and Odd Permutations Symmetries Conjugation Cayley's theorem 3 Groups of Symmetries 2 Permutations | every group is a Linear Isometries permutation group S n The Dihedral groups Representations and notations Symmetry Groups of the Permutation Statistics Platonic Solids A note on left vs right Transpositions, k-cycles, 4 Cayley's theorem | every generating sets group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman Summary The Symmetric group Permutations 1 The Symmetric group S2, S3, S4 Groups of Definition Even and Odd Permutations Symmetries Conjugation Cayley's theorem 3 Groups of Symmetries 2 Permutations | every group is a Linear Isometries permutation group S n The Dihedral groups Representations and notations Symmetry Groups of the Permutation Statistics Platonic Solids A note on left vs right Transpositions, k-cycles, 4 Cayley's theorem | every generating sets group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman Summary The Symmetric group Permutations 1 The Symmetric group S2, S3, S4 Groups of Definition Even and Odd Permutations Symmetries Conjugation Cayley's theorem 3 Groups of Symmetries 2 Permutations | every group is a Linear Isometries permutation group S n The Dihedral groups Representations and notations Symmetry Groups of the Permutation Statistics Platonic Solids A note on left vs right Transpositions, k-cycles, 4 Cayley's theorem | every generating sets group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman The Symmetric group Definition Conjugation Definition Permutations Let X be a set. Then the symmetric group on X is the group of all Groups of bijections Symmetries f : X X Cayley's theorem | every group is a with functional composition as operation. It is denoted by SX . permutation group ! Abstract Algebra, Lecture 5 Theorem Jan Snellman If X ; Y are sets, and φ : X Y is a bijection, then the maps The Symmetric group -1 Definition SX 3!f 7 φ ◦ f ◦ φ 2 SY Conjugation -1 Sy 3 g 7 φ ◦ g ◦ φ 2 SX Permutations ! Groups of are each other's inverses; thus, they are bijections. Furthermore, these Symmetries ! assignments respect the group operations of Sx and Sy , showing these two Cayley's theorem ~ -1 | every group is a groups to be isomorphic. We say that f and f = φ ◦ f ◦ φ are conjugate. permutation group The following commutative diagram illustrates conjugation: X f X φ φ ~ Y f Y Abstract Algebra, Lecture 5 Jan Snellman Definition The Symmetric For any positive integer n, we let [n] = f1; 2;:::; ng. We define the group symmetric group on n letters as Sn = S[n]. Permutations Sn Representations and Lemma notations Permutation Statistics If X is a set with n elements, then SX ' Sn. A note on left vs right Transpositions, k-cycles, generating sets Proof. S2, S3, S4 Number the elements in X to get a bijection φ :[n] X . Then the Even and Odd Permutations desired isomorphism is the conjugation Groups of ! Symmetries -1 SX 3 f 7 φ ◦ f ◦ φ 2 Sn Cayley's theorem | every group is a permutation group ! Abstract Algebra, Lecture 5 Jan Snellman Representations of permutations The Symmetric group Permutations • Let σ 2 Sn Sn 2 Representations and • Since σ :[n] [n], we can consider its graph, a subset of [n] notations Permutation Statistics 1 2 ··· n • Can represent σ in two-row notation as A note on left vs right ! σ( ) σ( ) σ( ) Transpositions, 1 2 ··· n k-cycles, generating sets • One-row notation is [σ(1); σ(2); ··· ; σ(n)] S2, S3, S4 Even and Odd • The associated bipartite graph has vertex set two copies of [n], a left Permutations and a right set, and an edge from i on the left to σ(i) on the right Groups of Symmetries • The associated directed graph has vertex set [i] and a directed edge Cayley's theorem from i to σ(i) | every group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman Example The Symmetric group • σ = [2; 9; 5; 6; 3; 4; 7; 1; 8] Permutations 9 Sn Representations and 8 notations Permutation Statistics 7 A note on left vs right 6 Transpositions, k-cycles, generating 5 sets 4 S2, S3, S4 Even and Odd 3 Permutations 2 Groups of Symmetries 1 • 1 2 3 4 5 6 7 8 9 Cayley's theorem | every group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman Example The Symmetric • σ = [2; 9; 5; 6; 3; 4; 7; 1; 8] group Permutations Sn Representations and notations Permutation Statistics 1 11 A note on left vs right Transpositions, 2 12 k-cycles, generating sets 3 13 S2, S3, S4 4 14 Even and Odd Permutations 5 15 Groups of 6 16 Symmetries 7 17 Cayley's theorem 8 18 | every group is a 9 19 permutation group • Abstract Algebra, Lecture 5 Example Jan Snellman The Symmetric • σ = [2; 9; 5; 6; 3; 4; 7; 1; 8] group • Permutations Sn Representations and notations Permutation Statistics A note on left vs right Transpositions, k-cycles, generating sets 2 6 S2, S3, S4 Even and Odd 3 Permutations 9 Groups of 7 Symmetries 1 Cayley's theorem 5 | every group is a permutation group 8 4 Abstract Algebra, Lecture 5 Jan Snellman • To understand the multiplication, the following representation is The Symmetric useful group Permutations • The associated linear map is the map Fσ : V V , where V is the Sn free vector space (over Q or whatever) with basis e1;:::; en, and Representations and notations with Fσ(ei ) = eσ(i) ! Permutation Statistics A note on left vs right • The associated permutation matrix Mσ is the matrix of the Transpositions, k-cycles, generating aforementioned map w.r.t. the natural ordered basis (the matrix acts sets on column vectors from the left) S2, S3, S4 Even and Odd • Permutations If τ is another permutation in Sn then Groups of Symmetries Fστ = Fσ ◦ Fτ Cayley's theorem Mστ = MσMτ | every group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman Example The Symmetric group • σ = [2; 9; 5; 6; 3; 4; 7; 1; 8] Permutations 0 1 Sn 0 0 0 0 0 0 0 1 0 Representations and B1 0 0 0 0 0 0 0 0C notations B C Permutation Statistics B C B0 0 0 0 1 0 0 0 0C A note on left vs right B C Transpositions, B0 0 0 0 0 1 0 0 0C k-cycles, generating B C sets • Mσ = B0 0 1 0 0 0 0 0 0C B C S2, S3, S4 B0 0 0 1 0 0 0 0 0C Even and Odd B C Permutations B C B0 0 0 0 0 0 0 0 0C Groups of B0 0 0 0 0 0 1 0 0C Symmetries @ A 0 1 0 0 0 0 0 0 1 Cayley's theorem | every group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman The Symmetric • In the directed graph each vertex has in-degree one and out-degree group one Permutations • Sn Hence the digraph decomposes into directed cycles Representations and notations • The cycle notation of σ is obtained by listing the elements of each Permutation Statistics cycles, as A note on left vs right Transpositions, (a1;:::; a` )(a` +1;:::; a` ) ··· () k-cycles, generating 1 1 2 sets S2, S3, S4 where, in each cycle, σ(av ) = av+1 Even and Odd Permutations • The cycle type of σ is the numerical partition of n given by the cycle Groups of lengths. It can be encoded in various ways; we'll usually write it as a Symmetries (non-strictly) decreasing sequence (c1; c2;:::; ck ) of cycle lengths. Cayley's theorem | every group is a permutation group Abstract Algebra, Lecture 5 Jan Snellman • σ = [2; 9; 5; 6; 3; 4; 7; 1; 8] • The Symmetric group Permutations Sn Representations and notations Permutation Statistics A note on left vs right 9 5 Transpositions, k-cycles, generating 4 sets 8 S2, S3, S4 7 Even and Odd 2 Permutations 6 Groups of 1 3 Symmetries Cayley's theorem | every group is a • σ = (1; 2; 9; 8)(3; 5)(4; 6)(7) permutation group • Cycle type is 9 = 4 + 2 + 2 + 1, written as (4; 2; 2; 1). Abstract Algebra, Lecture 5 Jan Snellman The Symmetric group Theorem Permutations σ, τ 2 Sn are conjugate if and only if they have the same cycle type. Sn Representations and notations Proof. Permutation Statistics A note on left vs right Order the cycles in order of decreasing length, breaking ties arbitrarily. Let Transpositions, k-cycles, generating φ be the bijection that associates the elements in correspoding cycles (in sets S2, S3, S4 cyclic order, picking a starting element in each cycle howsoever).