DOCSLIB.ORG

6 Permutation Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
6 Permutation Groups Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 6 Permutation Groups 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 permu- tation 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.
Load more

Recommended publications
  • On Some Generation Methods of Finite Simple Groups
    Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography On Some Generation Methods of Finite Simple Groups Ayoub B. M. Basheer Department of Mathematical Sciences, North-West University (Mafikeng), P Bag X2046, Mmabatho 2735, South Africa Groups St Andrews 2017 in Birmingham, School of Mathematics, University of Birmingham, United Kingdom 11th of August 2017 Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Abstract In this talk we consider some methods of generating finite simple groups with the focus on ranks of classes, (p; q; r)-generation and spread (exact) of finite simple groups. We show some examples of results that were established by the author and his supervisor, Professor J. Moori on generations of some finite simple groups. Ayoub Basheer, North-West University, South Africa Groups St Andrews 2017 Talk in Birmingham Introduction Preliminaries Special Kind of Generation of Finite Simple Groups The Bibliography Introduction Generation of finite groups by suitable subsets is of great interest and has many applications to groups and their representations. For example, Di Martino and et al. [39] established a useful connection between generation of groups by conjugate elements and the existence of elements representable by almost cyclic matrices. Their motivation was to study irreducible projective representations of the sporadic simple groups. In view of applications, it is often important to exhibit generating pairs of some special kind, such as generators carrying a geometric meaning, generators of some prescribed order, generators that offer an economical presentation of the group.
    [Show full text]
  • 4. Groups of Permutations 1
    4. Groups of permutations 1 4. Groups of permutations Consider a set of n distinguishable objects, fB1;B2;B3;:::;Bng. These may be arranged in n! different ways, called permutations of the set. Permu- tations can also be thought of as transformations of a given ordering of the set into other orderings. A convenient notation for specifying a given permutation operation is 1 2 3 : : : n ! , a1 a2 a3 : : : an where the numbers fa1; a2; a3; : : : ; ang are the numbers f1; 2; 3; : : : ; ng in some order. This operation can be interpreted in two different ways, as follows. Interpretation 1: The object in position 1 in the initial ordering is moved to position a1, the object in position 2 to position a2,. , the object in position n to position an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the positions of objects in the ordered set. Interpretation 2: The object labeled 1 is replaced by the object labeled a1, the object labeled 2 by the object labeled a2,. , the object labeled n by the object labeled an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the labels of the objects in the ABCD ! set. The labels need not be numerical { for instance, DCAB is a well-defined permutation which changes BDCA, for example, into CBAD. Either of these interpretations is acceptable, but one interpretation must be used consistently in any application. The particular application may dictate which is the appropriate interpretation to use. Note that, in either interpre- tation, the order of the columns in the permutation symbol is irrelevant { the columns may be written in any order without affecting the result, provided each column is kept intact.
    [Show full text]
  • 14.2 Covers of Symmetric and Alternating Groups
    Remark 206 In terms of Galois cohomology, there an exact sequence of alge- braic groups (over an algebrically closed field) 1 → GL1 → ΓV → OV → 1 We do not necessarily get an exact sequence when taking values in some subfield. If 1 → A → B → C → 1 is exact, 1 → A(K) → B(K) → C(K) is exact, but the map on the right need not be surjective. Instead what we get is 1 → H0(Gal(K¯ /K), A) → H0(Gal(K¯ /K), B) → H0(Gal(K¯ /K), C) → → H1(Gal(K¯ /K), A) → ··· 1 1 It turns out that H (Gal(K¯ /K), GL1) = 1. However, H (Gal(K¯ /K), ±1) = K×/(K×)2. So from 1 → GL1 → ΓV → OV → 1 we get × 1 1 → K → ΓV (K) → OV (K) → 1 = H (Gal(K¯ /K), GL1) However, taking 1 → µ2 → SpinV → SOV → 1 we get N × × 2 1 ¯ 1 → ±1 → SpinV (K) → SOV (K) −→ K /(K ) = H (K/K, µ2) so the non-surjectivity of N is some kind of higher Galois cohomology. Warning 207 SpinV → SOV is onto as a map of ALGEBRAIC GROUPS, but SpinV (K) → SOV (K) need NOT be onto. Example 208 Take O3(R) =∼ SO3(R)×{±1} as 3 is odd (in general O2n+1(R) =∼ SO2n+1(R) × {±1}). So we have a sequence 1 → ±1 → Spin3(R) → SO3(R) → 1. 0 × Notice that Spin3(R) ⊆ C3 (R) =∼ H, so Spin3(R) ⊆ H , and in fact we saw that it is S3. 14.2 Covers of symmetric and alternating groups The symmetric group on n letter can be embedded in the obvious way in On(R) as permutations of coordinates.
    [Show full text]
  • PERMUTATION GROUPS (16W5087)
    PERMUTATION GROUPS (16w5087) Cheryl E. Praeger (University of Western Australia), Katrin Tent (University of Munster),¨ Donna Testerman (Ecole Polytechnique Fed´ erale´ de Lausanne), George Willis (University of Newcastle) November 13 - November 18, 2016 1 Overview of the Field Permutation groups are a mathematical approach to analysing structures by studying the rearrangements of the elements of the structure that preserve it. Finite permutation groups are primarily understood through combinatorial methods, while concepts from logic and topology come to the fore when studying infinite per- mutation groups. These two branches of permutation group theory are not completely independent however because techniques from algebra and geometry may be applied to both, and ideas transfer from one branch to the other. Our workshop brought together researchers on both finite and infinite permutation groups; the participants shared techniques and expertise and explored new avenues of study. The permutation groups act on discrete structures such as graphs or incidence geometries comprising points and lines, and many of the same intuitions apply whether the structure is finite or infinite. Matrix groups are also an important source of both finite and infinite permutation groups, and the constructions used to produce the geometries on which they act in each case are similar. Counting arguments are important when studying finite permutation groups but cannot be used to the same effect with infinite permutation groups. Progress comes instead by using techniques from model theory and descriptive set theory. Another approach to infinite permutation groups is through topology and approx- imation. In this approach, the infinite permutation group is locally the limit of finite permutation groups and results from the finite theory may be brought to bear.
    [Show full text]
  • Binding Groups, Permutation Groups and Modules of Finite Morley Rank Alexandre Borovik, Adrien Deloro
    Binding groups, permutation groups and modules of finite Morley rank Alexandre Borovik, Adrien Deloro To cite this version: Alexandre Borovik, Adrien Deloro. Binding groups, permutation groups and modules of finite Morley rank. 2019. hal-02281140 HAL Id: hal-02281140 https://hal.archives-ouvertes.fr/hal-02281140 Preprint submitted on 9 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. BINDING GROUPS, PERMUTATION GROUPS AND MODULES OF FINITE MORLEY RANK ALEXANDRE BOROVIK AND ADRIEN DELORO If one has (or if many people have) spent decades classifying certain objects, one is apt to forget just why one started the project in the first place. Predictions of the death of group theory in 1980 were the pronouncements of just such amne- siacs. [68, p. 4] Contents 1. Introduction and background 2 1.1. The classification programme 2 1.2. Concrete groups of finite Morley rank 3 2. Binding groups and bases 4 2.1. Binding groups 4 2.2. Bases and parametrisation 5 2.3. Sharp bounds for bases 6 3. Permutation groups of finite Morley rank 7 3.1. Highly generically multiply transitive groups 7 3.2.
    [Show full text]
  • Representations of the Infinite Symmetric Group
    Representations of the infinite symmetric group Alexei Borodin1 Grigori Olshanski2 Version of February 26, 2016 1 Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia 2 Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow, Russia National Research University Higher School of Economics, Moscow, Russia Contents 0 Introduction 4 I Symmetric functions and Thoma's theorem 12 1 Symmetric group representations 12 Exercises . 19 2 Theory of symmetric functions 20 Exercises . 31 3 Coherent systems on the Young graph 37 The infinite symmetric group and the Young graph . 37 Coherent systems . 39 The Thoma simplex . 41 1 CONTENTS 2 Integral representation of coherent systems and characters . 44 Exercises . 46 4 Extreme characters and Thoma's theorem 47 Thoma's theorem . 47 Multiplicativity . 49 Exercises . 52 5 Pascal graph and de Finetti's theorem 55 Exercises . 60 6 Relative dimension in Y 61 Relative dimension and shifted Schur polynomials . 61 The algebra of shifted symmetric functions . 65 Modified Frobenius coordinates . 66 The embedding Yn ! Ω and asymptotic bounds . 68 Integral representation of coherent systems: proof . 71 The Vershik{Kerov theorem . 74 Exercises . 76 7 Boundaries and Gibbs measures on paths 82 The category B ............................. 82 Projective chains . 83 Graded graphs . 86 Gibbs measures . 88 Examples of path spaces for branching graphs . 91 The Martin boundary and Vershik{Kerov's ergodic theorem . 92 Exercises . 94 II Unitary representations 98 8 Preliminaries and Gelfand pairs 98 Exercises . 108 9 Spherical type representations 111 10 Realization of spherical representations 118 Exercises .
    [Show full text]
  • CS E6204 Lecture 5 Automorphisms
    CS E6204 Lecture 5 Automorphisms Abstract An automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges. Under composition, the set of automorphisms of a graph forms what algbraists call a group. In this section, graphs are assumed to be simple. 1. The Automorphism Group 2. Graphs with Given Group 3. Groups of Graph Products 4. Transitivity 5. s-Regularity and s-Transitivity 6. Graphical Regular Representations 7. Primitivity 8. More Automorphisms of Infinite Graphs * This lecture is based on chapter [?] contributed by Mark E. Watkins of Syracuse University to the Handbook of Graph The- ory. 1 1 The Automorphism Group Given a graph X, a permutation α of V (X) is an automor- phism of X if for all u; v 2 V (X) fu; vg 2 E(X) , fα(u); α(v)g 2 E(X) The set of all automorphisms of a graph X, under the operation of composition of functions, forms a subgroup of the symmetric group on V (X) called the automorphism group of X, and it is denoted Aut(X). Figure 1: Example 2.2.3 from GTAIA. Notation The identity of any permutation group is denoted by ι. (JG) Thus, Watkins would write ι instead of λ0 in Figure 1. 2 Rigidity and Orbits A graph is asymmetric if the identity ι is its only automor- phism. Synonym: rigid. Figure 2: The smallest rigid graph. (JG) The orbit of a vertex v in a graph G is the set of all vertices α(v) such that α is an automorphism of G.
    [Show full text]
  • Quasi P Or Not Quasi P? That Is the Question
    Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups.
    [Show full text]
  • Lecture 9: Permutation Groups
    Lecture 9: Permutation Groups Prof. Dr. Ali Bülent EKIN· Doç. Dr. Elif TAN Ankara University Ali Bülent Ekin, Elif Tan (Ankara University) Permutation Groups 1 / 14 Permutation Groups Definition A permutation of a nonempty set A is a function s : A A that is one-to-one and onto. In other words, a pemutation of a set! is a rearrangement of the elements of the set. Theorem Let A be a nonempty set and let SA be the collection of all permutations of A. Then (SA, ) is a group, where is the function composition operation. The identity element of (SA, ) is the identity permutation i : A A, i (a) = a. ! 1 The inverse element of s is the permutation s such that 1 1 ss (a) = s s (a) = i (a) . Ali Bülent Ekin, Elif Tan (Ankara University) Permutation Groups 2 / 14 Permutation Groups Definition The group (SA, ) is called a permutation group on A. We will focus on permutation groups on finite sets. Definition Let In = 1, 2, ... , n , n 1 and let Sn be the set of all permutations on f g ≥ In.The group (Sn, ) is called the symmetric group on In. Let s be a permutation on In. It is convenient to use the following two-row notation: 1 2 n s = ··· s (1) s (2) s (n) ··· Ali Bülent Ekin, Elif Tan (Ankara University) Permutation Groups 3 / 14 Symmetric Groups 1 2 3 4 1 2 3 4 Example: Let f = and g = . Then 1 3 4 2 4 3 2 1 1 2 3 4 1 2 3 4 1 2 3 4 f g = = 1 3 4 2 4 3 2 1 2 4 3 1 1 2 3 4 1 2 3 4 1 2 3 4 g f = = 4 3 2 1 1 3 4 2 4 2 1 3 which shows that f g = g f .
    [Show full text]
  • The Rotation Group of Rubik's Cube
    The Rotation Group of Rubik's Cube Oliver Knill Roman E. Mdder ETH Zurich Revised by Robin S. Sanders and Julie D. Simon University of Illinois, at Urbana-Champaign ABSTRACT In this project, the group theory program "Cayley" is used to solve some problems asso ciated with Rubik's Cube, including the original problem of restoring the initial state of the cube. Project-No.: 003-021 Title: Rubik's Cube Subject: Algebra Programs: Cayley References: 001-021 Die Drehgruppe des Rubikwurfels fypesetting: mltroff Version: 5/15/87 33 T h e M a t h e m a t i c a l L a b o r a t o r y R u b i k ' s C u b e 1. Introduction Several years have passed since the Rubik's Cube conquered the world. Almost everyone knows this cube and many have solved it or have tried to do so for hours. Rarely has there been a puzzle which has caused such a stir and which has found its way not only into the world of games but also into that of mathematics and computers. Before you start your work on this project about Rubik's cube, we recommend that you reach for a cube and get familiar with it (again). Many concepts of group theory can be demonstrated with this cube, and abstract definitions can become more intuitive. 1.1. The Cube If you already know it or are even a cube expert, you may skim this section which defines the structure of the cube and also introduces some notation.
    [Show full text]
  • Weyl Group Representations and Unitarity of Spherical Representations
    Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 ν1 = ν2 β S S ν α β Sβν Sαν ν SO(3,2) 0 α SαSβSαSβν = SβSαSβν SβSαSβSαν S S SαSβSαν β αν 0 1 2 ν = 0 [type B2] 2 1 0: Introduction Spherical unitary dual of real split semisimple Lie groups spherical equivalence classes of unitary-dual = irreducible unitary = ? of G spherical repr.s of G aim of the talk Show how to compute this set using the Weyl group 2 0: Introduction Plan of the talk ² Preliminary notions: root system of a split real Lie group ² De¯ne the unitary dual ² Examples (¯nite and compact groups) ² Spherical unitary dual of non-compact groups ² Petite K-types ² Real and p-adic groups: a comparison of unitary duals ² The example of Sp(4) ² Conclusions 3 1: Preliminary Notions Lie groups Lie Groups A Lie group G is a group with a smooth manifold structure, such that the product and the inversion are smooth maps Examples: ² the symmetryc group Sn=fbijections on f1; 2; : : : ; ngg à finite ² the unit circle S1 = fz 2 C: kzk = 1g à compact ² SL(2; R)=fA 2 M(2; R): det A = 1g à non-compact 4 1: Preliminary Notions Root systems Root Systems Let V ' Rn and let h; i be an inner product on V . If v 2 V -f0g, let hv;wi σv : w 7! w ¡ 2 hv;vi v be the reflection through the plane perpendicular to v. A root system for V is a ¯nite subset R of V such that ² R spans V , and 0 2= R ² if ® 2 R, then §® are the only multiples of ® in R h®;¯i ² if ®; ¯ 2 R, then 2 h®;®i 2 Z ² if ®; ¯ 2 R, then σ®(¯) 2 R A1xA1 A2 B2 C2 G2 5 1: Preliminary Notions Root systems Simple roots Let V be an n-dim.l vector space and let R be a root system for V .
    [Show full text]
  • 13. Symmetric Groups
    13. Symmetric groups 13.1 Cycles, disjoint cycle decompositions 13.2 Adjacent transpositions 13.3 Worked examples 1. Cycles, disjoint cycle decompositions The symmetric group Sn is the group of bijections of f1; : : : ; ng to itself, also called permutations of n things. A standard notation for the permutation that sends i −! `i is 1 2 3 : : : n `1 `2 `3 : : : `n Under composition of mappings, the permutations of f1; : : : ; ng is a group. The fixed points of a permutation f are the elements i 2 f1; 2; : : : ; ng such that f(i) = i. A k-cycle is a permutation of the form f(`1) = `2 f(`2) = `3 : : : f(`k−1) = `k and f(`k) = `1 for distinct `1; : : : ; `k among f1; : : : ; ng, and f(i) = i for i not among the `j. There is standard notation for this cycle: (`1 `2 `3 : : : `k) Note that the same cycle can be written several ways, by cyclically permuting the `j: for example, it also can be written as (`2 `3 : : : `k `1) or (`3 `4 : : : `k `1 `2) Two cycles are disjoint when the respective sets of indices properly moved are disjoint. That is, cycles 0 0 0 0 0 0 0 (`1 `2 `3 : : : `k) and (`1 `2 `3 : : : `k0 ) are disjoint when the sets f`1; `2; : : : ; `kg and f`1; `2; : : : ; `k0 g are disjoint. [1.0.1] Theorem: Every permutation is uniquely expressible as a product of disjoint cycles. 191 192 Symmetric groups Proof: Given g 2 Sn, the cyclic subgroup hgi ⊂ Sn generated by g acts on the set X = f1; : : : ; ng and decomposes X into disjoint orbits i Ox = fg x : i 2 Zg for choices of orbit representatives x 2 X.
    [Show full text]