Abstract Algebra, Lecture 5

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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

Jan Snellman Summary

The Symmetric group

Jan Snellman Summary

The Symmetric group

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 ∈ SY Conjugation −1 Sy 3 g 7 φ ◦ g ◦ φ ∈ 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] = {1, 2,..., n}. 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 ∈ 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 ∈ 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   Sn 0 0 0 0 0 0 0 1 0 Representations and 1 0 0 0 0 0 0 0 0 notations   Permutation Statistics   0 0 0 0 1 0 0 0 0 A note on left vs right   Transpositions, 0 0 0 0 0 1 0 0 0 k-cycles, generating   sets • Mσ = 0 0 1 0 0 0 0 0 0   S2, S3, S4 0 0 0 1 0 0 0 0 0 Even and Odd   Permutations   0 0 0 0 0 0 0 0 0 Groups of 0 0 0 0 0 0 1 0 0 Symmetries   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 ∈ 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). Even and Odd Permutations Conjugate using φ. Groups of Symmetries Note that in this case we have that X = Y = [n], and that φ ∈ Sn, as well. Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman Example

The Symmetric group Permutations σ = (1, 2, 9, 8)(3, 5)(4, 6)(7) Sn τ = (2, 3, 4, 5)(6, 7)(8, 9)(1) Representations and notations Permutation Statistics are conjugate, for instance using φ = A note on left vs right Transpositions, k-cycles, generating 1 2 3 4 5 6 7 8 9 sets S2, S3, S4 2 3 6 8 7 9 1 5 4 Even and Odd Permutations Check that Groups of −1 Symmetries τ = φσφ , Cayley’s theorem or is it — every group is a −1 permutation group τ = φ σφ? Abstract Algebra, Lecture 5

Jan Snellman Deﬁnition The Symmetric group Let Sn 3 σ = [σ(1), . . . , σ(n)]. Permutations • A descent is an index 1 ≤ i < n such that σ(i) > σ(i + 1). We use Sn Representations and Des(σ) ⊆ [n − 1] for the set of descents, and des(σ) for the number notations maj Permutation Statistics of descents. The major index (σ) is the sum of the elements in A note on left vs right the descent set. Transpositions, k-cycles, generating sets • An inversion is a pair of indices 1 ≤ i < j ≤ n such that σ(i) > σ(j). 2 S2, S3, S4 We use Inv(σ) ⊆ [n] for the set of inversions, and inv(σ) for the Even and Odd Permutations number of inversions. Groups of • There are many, many more permutation statistics. Their Symmetries enumeration is a huge topic in algebra and combinatorics! Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric Example group Permutations • σ = [2, 9, 5, 6, 3, 4, 7, 1, 8] • Descent set is 2, 4, 7 Sn Representations and 9 • Inversion set is notations 8

Permutation Statistics 7 (2, 3), (2, 4), (2, 5), (2, 6), A note on left vs right 6 Transpositions, (2, 7), (2, 8), (2, 9), (4, 5), k-cycles, generating 5 sets (4, 6), (4, 8), (6, 8), (7, 8). 4 S2, S3, S4 3 There are thus 12 inversions. Even and Odd Permutations 2 Groups of 1 Symmetries • 1 2 3 4 5 6 7 8 9 Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman Left versus right

The Symmetric group Permutations • We (I, Judson, Svensson) write f (x) for a function

Sn Representations and • Furthermore, (g ◦ f )(x) = g(f (x)) notations Permutation Statistics • Arguments enter from the right, and gets transported leftwards A note on left vs right Transpositions, • In particular, if f , g ∈ Sn, then fg ∈ Sn is the bijection “ﬁrst apply g, k-cycles, generating sets then apply f to the result” S2, S3, S4 Even and Odd • Other authors prefer to write (x)f or just xf . Permutations • Then fg means “ﬁrst apply f, then apply g to the result” Groups of Symmetries • In particular, SAGE (and GAP) uses this convention Cayley’s theorem • Do not become confused! — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman Deﬁnition The Symmetric group • A k-cycle is a permutation whose cycle type has a single cycle of Permutations length k, and possibly a bunch of cycles of length one.

Sn Representations and • For instance, (1, 2, 3)(4)(5) ∈ S5 is a 3-cycle. notations Permutation Statistics • A transposition is another name for a 2-cycle. A note on left vs right Transpositions, • An involution is a permutation whose square is the identity k-cycles, generating sets • The disjoint cycle factorization is the factorization of the permutation S2, S3, S4 Even and Odd into cycles as given by the cycle notation Permutations • For instance, if f = (1, 2, 3)(4, 5, 6) ∈ S6 then we can write f = gh Groups of Symmetries with g = (1, 2, 3)(4)(5)(6) and h = (4, 5, 6)(1)(2)(3) Cayley’s theorem • It is common and convenient to omit the cycles of length one. — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Lemma Permutations Sn • Disjoint cycles commute Representations and notations • Transpositions are involutions Permutation Statistics A note on left vs right • Involutions are products of disjoint transpositions, hence have cycle Transpositions, k-cycles, generating type (2, 2, 2,..., 2, 1,..., 1). sets S2, S3, S4 • A k-cycle has order k. Even and Odd Permutations • The order of a permutation is the l.c.m. of the sizes of its cycles Groups of Symmetries Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5 Example Jan Snellman If σ = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) then its powers have cycle structure as follows: The Symmetric group 0 () Permutations 1 (1,2,3,4,5,6,7,8,9,10,11,12) Sn 2 (1,3,5,7,9,11)(2,4,6,8,10,12) Representations and notations 3 (1,4,7,10)(2,5,8,11)(3,6,9,12) Permutation Statistics 4 (1,5,9)(2,6,10)(3,7,11)(4,8,12) A note on left vs right Transpositions, 5 (1,6,11,4,9,2,7,12,5,10,3,8) k-cycles, generating 6 (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) sets S2, S3, S4 7 (1,8,3,10,5,12,7,2,9,4,11,6) Even and Odd 8 (1,9,5)(2,10,6)(3,11,7)(4,12,8) Permutations 9 (1,10,7,4)(2,11,8,5)(3,12,9,6) Groups of 10 (1,11,9,7,5,3)(2,12,10,8,6,4) Symmetries 11 (1,12,11,10,9,8,7,6,5,4,3,2) Cayley’s theorem 12 () — every group is a permutation group In general, the k-th power of an m-cycle consists of (m/ gcd(k, m))-cycles, and gcd(k, m) such cycles. Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Permutations Example Sn Representations and notations Non-disjoint cycles need not commute: Permutation Statistics A note on left vs right (1, 2)(1, 3) = (1, 3, 2) Transpositions, k-cycles, generating sets (1, 3)(1, 2) = (1, 2, 3) S2, S3, S4 Even and Odd Permutations Groups of Symmetries Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5 Example Jan Snellman Let σ = (1, 2, 3, 4)(5, 6, 7)(8, 9, 10). Then o(σ) = lcm(4, 3, 3) = 12, and The Symmetric its powers are group Permutations 0 () Sn 1 (1,2,3,4)(5,6,7)(8,9,10) Representations and notations 2 (1,3)(2,4)(5,7,6)(8,10,9) Permutation Statistics A note on left vs right 3 (1,4,3,2) Transpositions, k-cycles, generating 4 (5,6,7)(8,9,10) sets 5 (1,2,3,4)(5,7,6)(8,10,9) S2, S3, S4 Even and Odd 6 (1,3)(2,4) Permutations 7 (1,4,3,2)(5,6,7)(8,9,10) Groups of Symmetries 8 (5,7,6)(8,10,9) Cayley’s theorem 9 (1,2,3,4) — every group is a 10 (1,3)(2,4)(5,6,7)(8,9,10) permutation group 11 (1,4,3,2)(5,7,6)(8,10,9) 12 () Abstract Algebra, Lecture 5 Jan Snellman Lemma

The Symmetric Any k-cycle can be written as a product of transpositions. group Permutations Thus, since every permutation is the product of disjoint cycles, the Sn transpositions generate Sn. Representations and notations Permutation Statistics Proof. A note on left vs right Transpositions, It is enough to note that k-cycles, generating sets S2, S3, S4 (1, 2, 3,..., k) = (1, 2) ··· (k − 2, k − 1)(k − 1, k) Even and Odd Permutations Groups of Symmetries Example Cayley’s theorem — every group is a (1,2)(2,3)(3,4) = (1,2,3,4) permutation group Abstract Algebra, Lecture 5 Jan Snellman Lemma

The Symmetric Every transposition is the product of an odd number of adjacent group transpositions, i.e. transpositions of the form (j, j + 1) Permutations S n So Sn is generated by adjacent transpositions. Representations and notations Permutation Statistics Proof. A note on left vs right Transpositions, Enough to note that k-cycles, generating sets S2, S3, S4 (1, k − 1)(k − 1, k)(1, k − 1) = (1, k) Even and Odd Permutations Groups of Symmetries Example Cayley’s theorem — every group is a permutation group (1,4) = (1,3)(3,4)(1,3) = (1,2)(2,3)(1,2)(3,4)(1,2)(2,3)(1,2) Abstract Algebra, Lecture 5

Jan Snellman Lemma

The Symmetric Sn is generated by {(1, 2), (1, 2, 3,..., n)} group Permutations So, while not cyclic, Sn has a generating set with just two elements.

Sn Representations and Proof. notations Permutation Statistics Enough to note that A note on left vs right Transpositions, j −j k-cycles, generating (j + 1, j + 2) = (1, 2, 3,..., n) (1, 2)(1, 2, 3,..., n) sets S2, S3, S4 Even and Odd Permutations Groups of Example Symmetries Cayley’s theorem — every group is a (1, 2, 3, 4)2(1, 2)(1, 2, 3, 4)−2 = (1)(2)(3, 4) permutation group Abstract Algebra, Lecture 5

Jan Snellman S2

The Symmetric group Permutations

Sn Representations and notations Example Permutation Statistics A note on left vs right S2 is cyclic: Transpositions, ∗ () (1, 2) k-cycles, generating sets () () (1, 2) S2, S3, S4 Even and Odd (1, 2) (1, 2) () Permutations Groups of Symmetries Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman S3

The Symmetric group Permutations Example

Sn Representations and S3 is the smallest non-abelian group: notations Permutation Statistics ∗ () (1, 3, 2)(1, 2, 3)(2, 3)(1, 3)(1, 2) A note on left vs right Transpositions, () () (1, 3, 2)(1, 2, 3)(2, 3)(1, 3)(1, 2) k-cycles, generating sets (1, 3, 2) (1, 3, 2)(1, 2, 3) () (1, 2)(2, 3)(1, 3) S2, S3, S4 (1, 2, 3) (1, 2, 3) () (1, 3, 2)(1, 3)(1, 2)(2, 3) Even and Odd Permutations (2, 3) (2, 3)(1, 3)(1, 2) () (1, 3, 2)(1, 2, 3) Groups of (1, 3) (1, 3)(1, 2)(2, 3)(1, 2, 3) () (1, 3, 2) Symmetries (1, 2) (1, 2)(2, 3)(1, 3)(1, 3, 2)(1, 2, 3) () Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman S4

The Symmetric Example group S4 has 4! = 24 elements, with the following orders: Permutations Sn () 1 (3,4) 2 Representations and notations (1,3)(2,4) 2 (1,3,2,4) 4 Permutation Statistics A note on left vs right (1,4)(2,3) 2 (1,4,2,3) 4 Transpositions, (1,2)(3,4) 2 (1,2) 2 k-cycles, generating sets (2,3,4) 3 (2,3) 2 S2, S3, S4 Even and Odd (1,3,2) 3 (1,3,4,2) 4 Permutations (1,4,3) 3 (1,4) 2 Groups of Symmetries (1,2,4) 3 (1,2,4,3) 4 Cayley’s theorem (2,4,3) 3 (2,4) 2 — every group is a (1,3,4) 3 (1,3) 2 permutation group (1,4,2) 3 (1,4,3,2) 4 (1,2,3) 3 (1,2,3,4) 4 Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Permutations Deﬁnition Sn Representations and notations The sign of a permutation σ ∈ Sn is Permutation Statistics A note on left vs right sgn(σ) = (−1)inv(σ). Transpositions, k-cycles, generating sets Permutations with sign +1 are even, the rest are odd. S2, S3, S4 Even and Odd Permutations Groups of Symmetries Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5 Lemma Jan Snellman sgn((i, i + 1)) = −1. The Symmetric group Proof. Permutations Inv Sn We have that ((i, i + 1)) = {(i, i + 1)}. Representations and notations Permutation Statistics Lemma A note on left vs right Transpositions, Any transposition is odd. k-cycles, generating sets S2, S3, S4 Proof. Even and Odd Permutations If τ = (a, b) with a < b then Inv(τ) contains Groups of Symmetries • (a, c) for a < c < b, Cayley’s theorem • (c, b) for a < c < b, and — every group is a permutation group • (a, b). This is an odd number. Abstract Algebra, Lecture 5

Jan Snellman The only hard proof today — proof skipped

The Symmetric group Permutations

Sn Representations and Theorem notations Permutation Statistics If σ, τ ∈ Sn, and τ is a transposition, then |inv(σ) − inv(τσ)| is odd. A note on left vs right Transpositions, k-cycles, generating sets Proof. S2, S3, S4 Even and Odd You absolutely, unequivocally, positively, have to read this proof in your Permutations textbook! (It is a case-by-case study, similar to the previous lemma) Groups of Symmetries Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman Example The Symmetric group 1 σ = [2, 9, 5, 6, 3, 4, 7, 1, 8] 3 (3, 7)σ = [2, 9, 5, 6, 7, 4, 3, 1, 8] Permutations 9 9 Sn 8 8 Representations and notations 7 7 Permutation Statistics 6 6 A note on left vs right 5 5 Transpositions, 4 4 k-cycles, generating 3 3 sets 2 2 S2, S3, S4 1 1 Even and Odd 2 1 2 3 4 5 6 7 8 9 4 1 2 3 4 5 6 7 8 9 Permutations Groups of New inversions: (5, 6), (5, 7), (5, 8) Symmetries Disappearing inversions: (3, 5), (4, 5) Cayley’s theorem — every group is a Net gain: +1 inversions permutation group Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Corollary Permutations Let σ ∈ Sn. Then, if Sn m Representations and notations σ = τj Permutation Statistics j=1 A note on left vs right Y Transpositions, is a factorization of σ as a product of transpositions, then k-cycles, generating sets S2, S3, S4 (−1)m = sgn(σ), Even and Odd Permutations Groups of that is, m is odd if σ is an odd permutation, and even if σ is even. Symmetries Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman Corollary The Symmetric group If σ, γ ∈ Sn, then sgn sgn sgn Permutations (σγ) = (σ) (γ).

Sn Representations and notations Proof. Permutation Statistics A note on left vs right Write σ and γ as a product of transpositions. Transpositions, k-cycles, generating sets S2, S3, S4 Corollary Even and Odd Permutations The determinant of Mσ is the sign of σ. Groups of Symmetries Proof. Cayley’s theorem — every group is a It is true for transpositions. permutation group Abstract Algebra, Lecture 5

Jan Snellman Deﬁnition The Symmetric group Let E n be the n-dimensional Euclidean space, with the standard inner Permutations product. A linear isometry is a linear map F : E n E n preserving the Groups of inner product, i.e, Symmetries hF (u), F (v)i = hu, vi Linear Isometries → The Dihedral groups n Symmetry Groups of for all vectors u, v ∈ E . the Platonic Solids Cayley’s theorem Clearly, F also preserves norms, and distances. — every group is a permutation group Deﬁnition We denote the group of linear isometries of E n by LI(E n).

n This is a subgroup of SE n , the group of all bijections on E . Abstract Algebra, Lecture 5 Lemma Jan Snellman • The matrix M of the linear isometry F satisfies M ∗ Mt = I . It has The Symmetric group either determinant +1, if F is orientation-preserving, or −1, if F is Permutations orientation-reversing. Groups of • In the plane, the isometries are either the orientation-preserving Symmetries rotations, with matrix Linear Isometries The Dihedral groups Symmetry Groups of cos(φ) − sin(φ) the Platonic Solids sin(φ) cos(φ) Cayley’s theorem — every group is a permutation group or reflections in a line through the origin with (unit) normal vector w, given by u 7 u − 2hu, wiw • In E 3, there are orientation-preserving rotations, and orientation-reversing rotation-reflections→ Abstract Algebra, Lecture 5

Jan Snellman Deﬁnition 2 The Symmetric Let, for n ≥ 3, Kn be the regular n-gon in E , realized as the convex hull group of its set of vertices Permutations = ( ( π/ ), ( π/ )) < Groups of Vn { cos 2k n sin 2k n 0 ≤ k n } Symmetries Linear Isometries The Dihedral groups Symmetry Groups of Example the Platonic Solids

Cayley’s theorem 1 — every group is a 0.5 0.5 permutation group 0.5

-0.4 -0.2 0.2 0.4 0.6 0.8 1 -0.5 0.5 1 -1 -0.5 0.5 1

-0.5 -0.5 -0.5

-1 Abstract Algebra, Lecture 5

Jan Snellman Deﬁnition

The Symmetric The Dihedral group Dn is the group of symmetries of Kn, i.e. group Permutations n Dn = { F ∈ LI(E ) F (Kn) = Kn } Groups of Symmetries Linear Isometries The Dihedral groups Theorem Symmetry Groups of the Platonic Solids An element of Dn must restrict to a bijection on Vn. Two diﬀerent Cayley’s theorem elements of Dn induce diﬀerent permutations of Vn. Thus — every group is a permutation group LI n Dn = { F ∈ (E ) F (Vn) = Vn } ' G ≤ SVn ' Sn

In general, not every permutation of the vertices can be extended to a linear isometry preserving Kn. Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Theorem Permutations Dn has 2n elements; n rotations and n reflections. Groups of Symmetries Linear Isometries Proof. The Dihedral groups Symmetry Groups of Imagine cutting out Kn from the plane and then putting it back, filling the the Platonic Solids hole. You either flip the cut-out, or you don’t. Cayley’s theorem — every group is a permutation group We will label the vertices counter-clockwise, starting with (1, 0) as vertex number 1. Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Example

Permutations Any permutation of the vertices of the triangle K3 can be obtained by a Groups of rotation or a reﬂection. Thus D3 ' S3. Symmetries Linear Isometries The Dihedral groups Symmetry Groups of the Platonic Solids Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Example Permutations Since adjacent vertices of K4 must remain adjacent after a symmetry, the Groups of permutation (2, 3) is impossible. Symmetries Linear Isometries The Dihedral groups Symmetry Groups of the Platonic Solids Cayley’s theorem — every group is a permutation group In fact, D4 has multiplication table ∗ () (1, 3)(2, 4)(1, 4, 3, 2)(1, 2, 3, 4)(2, 4)(1, 3)(1, 4)(2, 3)(1, 2)(3, 4) () () (1, 3)(2, 4)(1, 4, 3, 2)(1, 2, 3, 4)(2, 4)(1, 3)(1, 4)(2, 3)(1, 2)(3, 4) (1, 3)(2, 4) (1, 3)(2, 4) () (1, 2, 3, 4)(1, 4, 3, 2)(1, 3)(2, 4)(1, 2)(3, 4)(1, 4)(2, 3) (1, 4, 3, 2) (1, 4, 3, 2)(1, 2, 3, 4)(1, 3)(2, 4) () (1, 2)(3, 4)(1, 4)(2, 3)(2, 4)(1, 3) (1, 2, 3, 4) (1, 2, 3, 4)(1, 4, 3, 2) () (1, 3)(2, 4)(1, 4)(2, 3)(1, 2)(3, 4)(1, 3)(2, 4) (2, 4) (2, 4)(1, 3)(1, 4)(2, 3)(1, 2)(3, 4) () (1, 3)(2, 4)(1, 4, 3, 2)(1, 2, 3, 4) (1, 3) (1, 3)(2, 4)(1, 2)(3, 4)(1, 4)(2, 3)(1, 3)(2, 4) () (1, 2, 3, 4)(1, 4, 3, 2) (1, 4)(2, 3) (1, 4)(2, 3)(1, 2)(3, 4)(1, 3)(2, 4)(1, 2, 3, 4)(1, 4, 3, 2) () (1, 3)(2, 4) (1, 2)(3, 4) (1, 2)(3, 4)(1, 4)(2, 3)(2, 4)(1, 3)(1, 4, 3, 2)(1, 2, 3, 4)(1, 3)(2, 4) () Abstract Algebra, Lecture 5 Jan Snellman Theorem

The Symmetric Let r ∈ Dn denote rotation by 2π/n radians counter-clockwise, and let s group denote reﬂection in the x-axis. Permutations 1 r induces the permutation (1, 2,..., n). It has order n. Groups of Symmetries 2 s induces (1)(2, n − 1) ··· ((n + 1)/2c, (n + 3)/2) if n is odd, and Linear Isometries (1)(2, n − 1) ··· (n/2, n/2 + 2)((n + 1)/2) if n is even. It has order The Dihedral groups Symmetry Groups of two. the Platonic Solids Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5 Theorem (Contd)

Jan Snellman 3 If n is odd, then the reflection in the line through the vertex k can be The Symmetric given as group r k−1sr 1−k Permutations If n is even, then the reflection in the line through the opposite vertices k Groups of and (n + 1)/2 + (k − 1) is given by Symmetries Linear Isometries r k−1sr 1−k , The Dihedral groups Symmetry Groups of the Platonic Solids whereas the reflection in the line through the midpoint between k and Cayley’s theorem k + 1 is given by — every group is a (k, k + 1)(k − 1, k + 2) ··· permutation group Abstract Algebra, Lecture 5

Jan Snellman Theorem (Contd) The Symmetric group 4 All rotations commute Permutations 4 The product of two reﬂections is a rotation (which?) Groups of Symmetries 4 Regardless of the parity of n, the following relation hold: Linear Isometries The Dihedral groups srs = r −1 Symmetry Groups of the Platonic Solids Cayley’s theorem 4 Dn = hr, si, and all relations can be derived from — every group is a permutation group r n = s2 = srsr = 1

a b 4 Thus, the elements of Dn can be listed as s r with 0 ≤ a ≤ 1, 0 ≤ b < n Abstract Algebra, Lecture 5 Theorem Jan Snellman Let G be the group of symmetries of the regular tetrahedron. Then G has The Symmetric group 4 ∗ 3 = 12 rotations, and equally many rotation-reﬂections. Thus G has 24 Permutations elements in total; since G permutes the four vertices of the tetrahedron, and 4! = 24, we must have that G ' S . The subgroup of rotations Groups of 4 Symmetries correspond to even permutations. Linear Isometries The Dihedral groups From Wikipedia: Symmetry Groups of the Platonic Solids Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5 Theorem Jan Snellman The symmetry group of the cube has 48 elements, of which half are The Symmetric rotations. The subgroup of rotations is isomorphic to S4. group Permutations Groups of Symmetries Linear Isometries The Dihedral groups Symmetry Groups of the Platonic Solids Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5 Theorem Jan Snellman The symmetry group of the cube has 48 elements, of which half are The Symmetric rotations. The subgroup of rotations is isomorphic to S4. group Permutations Groups of Symmetries Linear Isometries The Dihedral groups Symmetry Groups of the Platonic Solids Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5 Theorem Jan Snellman The symmetry group of the cube has 48 elements, of which half are The Symmetric rotations. The subgroup of rotations is isomorphic to S4. group Permutations Groups of Symmetries Linear Isometries The Dihedral groups Symmetry Groups of the Platonic Solids Cayley’s theorem — every group is a permutation group Abstract Algebra, Lecture 5

Jan Snellman Example The Symmetric group Consider again the symmetries of the tetrahedron Permutations Groups of Symmetries Cayley’s theorem — every group is a permutation group

• Any symmetry permutes the vertices. For instance, the ﬁrst rotation induces (1, 2, 3)(4), labeling the vertices on the blue triangle ﬁrst. • The same rotation induces (a, b, c)(d, e, f ) on the edges Abstract Algebra, Lecture 5 Example Jan Snellman

The Symmetric group Permutations Groups of Symmetries Cayley’s theorem — every group is a • The rotations of the cube acts on the four space diagonals, and each permutation group possible permutation of space diagonals can be so obtained. This is one way of showing that the rotations form a group isomorphic to S4 • The full isomorphism group of the cube has 48 elements. • Thus there is a pair of diﬀerent symmetries that permutes the space diagonals in the same way! • In fact, the antipodal map z 7 −z ﬁxes the space diagonals, just like the identity map z 7 z. → → Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric • There is a canonical way of representing group elements as group permutations, so that Permutations 1 Different group elements yields different permutations Groups of 2 Products of group elements correspond to products of their induced Symmetries permutations Cayley’s theorem • The name of this magnificent construction is... — every group is a permutation group • Cayley’s left regular representation!!!!! • Works even for infinite groups! • Is intuitive and straightforward — once learned, will never be forgotten! • Just make sure not to confuse left and right... Abstract Algebra, Lecture 5

Jan Snellman

Jan Snellman Injective homomorphisms

The Symmetric Deﬁnition group Let G, H be group. A map φ : G H is a homomorphism if for all Permutations x, y ∈ G, Groups of φ(xy)→ = φ(x)φ(y) Symmetries Cayley’s theorem — every group is a Lemma permutation group φ(1) = 1, and φ(g −1) = φ(g)−1.

Proof. Take x ∈ G. Then φ(x) = φ(1x) = φ(1)φ(x), so φ(1) acts as the identity. Consider φ(gg −1). On one hand, gg −1 = 1, and we have shown that φ(1) = 1. On the other hand, φ(gg −1) = φ(g)φ(g −1). Hence φ(g −1) acts as the inverse of φ(g). Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Permutations Lemma Groups of Symmetries • φ(G) ≤ H. Cayley’s theorem • φ−1(H) ≤ G, and is trivial iﬀ φ is injective. — every group is a permutation group • If φ is an injective homomorphism, then φ(G) ' G. • If G ≤ H then the inclusion is an injective homomorphism. Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric Example group Permutations The exponential map 3 k 7 exp(ki) ∈ T Groups of Z Symmetries is a homomorphism, by the functional equation of the complex exponential Cayley’s theorem → — every group is a function. It is injective, since permutation group k exp(ki) = 1 ∈ k = 0 2π Z

The image is the cyclic subgroup⇐⇒ of T generated⇐⇒ by exp(i); this subgroup is inﬁnite cyclic. The kernel is trivial. Abstract Algebra, Lecture 5

Jan Snellman Theorem (Cayley) The Symmetric group Let G be a group. Then there is a set X , and a subgroup H ≤ SX , such Permutations that G ' H. Groups of Symmetries Proof Cayley’s theorem — every group is a 1 Take any g ∈ G permutation group 2 Deﬁne vg : G G as left multiplication by g, so vg (x) = gx 3 From lemma “Linear equations in group have unique soln” follows that vg is a bijection→ (not an isomorphism though, does not respect multiplication).

4 So vg ∈ SG Abstract Algebra, Lecture 5

Jan Snellman

The Symmetric group Proof (contd) Permutations Groups of 5 vgh(x) = (gh) ∗ x = g ∗ (h ∗ x) = vg (vh(x)) = (vg ◦ vh)(x) Symmetries 6 So G 3 g 7 vg ∈ SG respects multiplication Cayley’s theorem — every group is a 7 Furthermore, g1 6= g2 = vg1 6= vg2 (evaluate at 1) permutation group → 8 So G 3 g 7 vg ∈ SG is an injective homomorphism. ⇒ 9 Call its image H; this is a subgroup of SX , with X = G. → 10 Furthermore G ' H. Abstract Algebra, Lecture 5

Jan Snellman Example The Symmetric group Let G be Klein’s Viergruppe, in its rows in the multiplication table): Permutations guise Groups of Symmetries * I S T R I − (I )(S)(T )(R) Cayley’s theorem I I S T R S − (I , S)(T , R) — every group is a → permutation group S S I R T T − (I , T )(S, R) T T R I S R −→ (I , R)(S, T ) R R T S I → Then left multiplication by the These four permutations→ form a various group elements induce the subgroup of S{I ,S,T ,R} ' S4 which is folowing permutations (read the isomorphic to G. Abstract Algebra, Lecture 5 Example Jan Snellman Let G = S3, and label the elements as a through f , inte the order The Symmetric [, (1, 3, 2), (1, 2, 3), (2, 3), (1, 3), (1, 2)] . The multiplication table is then group Permutations ∗ a b c d e f Groups of a a b c d e f Symmetries b b c a f d e Cayley’s theorem — every group is a c c a b e f d permutation group d d e f a b c e e f d c a b f f d e b c a

The group elements induce the permutations

(), (bca)(dfe), (acb)(def ), (ad)(be)(cf ), (ae)(bf )(cd), (af )(bd)(ce)

which form a subgroup of S{a,b,c,d,e,f } ' S6