DOCSLIB.ORG

Permutations and Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Permutations and Groups MATH 47A FALL 2008 INTRO TO MATH RESEARCH 1 1. Permutations or: group theory in 15 minutes For those of you who already took a course in group theory, you probably learned about “abstract groups” which are sets with binary operations satisfying a list of conditions. I want to talk about permu- tations. Those who already know group theory can think about the question: Why is every group isomorphic to a permutation group? Definition 1.1. A permutation on a set X is a bijection f : X X. Recall that a bijection is a mapping which is → (1) 1-1 (injective) and (2) onto (surjective). If X = 1, 2, , n , a permutation of X is called a permutation on n letters. {The set· · · of p}ermutations of X will be denoted by P erm(X). I discussed three notations for permutations: 1 2 3 4 (1) σ = means σ(1) = 3, σ(2) = 4, σ(3) = 2, σ(4) = 3 4 2 1 1 or σ!(x) = y where" x is given in the first row and y is given in the second row. (2) (cycle form) σ = (1324). This means σ sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1: 1 3 2 4 1. (3) (graphic notation).→Here→ you→ put→ the numbers x = 1, , n vertically on the right, put y = 1, , n vertically on the· · ·left and connect each x to each y = f· ·(x· ). For example, if f = (1423), you connect 1 on the right to f(1) = 4 on the left with a straight line, etc. In cycle notation, f = (1423). y x f x 1 3 M 1 f MM z MMM zz MM zz MMM zz MMM zz 2 4 MM zz 2 fMM MMz hhhh MM zzMMhhhh MMM zhzhhhMMM MM hhhzh MM hhhMMh zz MM hhhh zMzMM MM 3 2 thh zz MM 3 zz MMM zz MM zz MMM zz MMM zz MM 4 1 } 4 2 NOTES Definition 1.2. If f, g P erm(X), fg = f g is the composition of f and g. This is the perm∈ utation defined by ◦ fg(x) = f(g(x)) It means you do g first and then f. Question: If g = (34) what is gf? Write the answer in all three nota- tions and demonstrate the composition in the notation. (1) Stack up f, g, putting the first operation f on top and the second g underneath. Then cross out the second line: 1 2 3 4 1 2 3 4 4 3 1 2 = 3 4 1 2 3 4 1 2 ! " (2) Write the cycles next to each other and compute the image of each x by applying the cycles one at a time going from right to left: gf = (34)(1423) = (13)(24) for example: gf(1) =3 (34)4(1423)1 = 3 (3) Draw the diagrams next to each other, putting the first permu- tation f on the right: y g f x o 1 3 3 fMM 1 MM zz MMM zz MMM zz MM zz MMM zz 2 4 o 4 fM MM z hh 2 MM MzMz hhhh MMM zzhhMMhMh MM hzhzhh MM MhMhhhz MM hhhh MMzz MM thhhh zzMMM M 3 1 gN 2 zz MM 3 NNN ppp z MM NN pp zz MMM pNpNNp zz MM pp NN zz MMM ppp NNN zz MM 4 2 wp 1 } 4 Note that g = (34) switches the “letters” in locations 3 and 4. We discussed the fact that two of the crossings cancel when we redraw the picture: MATH 47A FALL 2008 INTRO TO MATH RESEARCH 3 y gf x 1 3 iSSS kk 1 SSS kkk SSS kkk SSS kkk SSS kkk SSS kkk SSS kkk 2 4 iSS kkkSSS k 2 SSS kkk SSS kkk SSS kkk SSS kkk SSSkkk SSkSkk kkk SSS kkk SSS kkk SSS kkk SSS kkk SSS kkk SSS 3 1 uk kSkSSk 3 kkk SSS kkk SSS kkk SSS kkk SSS kkk SSS kkk SSS 4 2 ukk S 4 Then we discussed the number of crossings. 4 NOTES 2. Transpositions On Day 2, I tried to formalize things we talked about at the end of Day 1, namely, in what way do the crossings in the diagram give transpositions and: What is the “longest word” ? Definition 2.1. A transposition is a 2-cycle (ab). For example 1 2 3 4 (14) = 4 2 3 1 ! " is a transposition. A simple transposition (also called a simple reflec- tion) is a transposition of two consecutive letters: si := (i, i + 1) In the group of permutations of n + 1 letters there are n simple reflec- tions: s = (12), s = (23), , s = (n, n + 1). 1 2 · · · n We talked about the following vaguely phrased theorem. Theorem 2.2. A permutation f can be written as a product of simple reflections, one for every crossing in the diagram for f. I drew a bunch of diagrams to illustrate this. The first was: y g x 1 3 fM hh 1 MM hhhh MM hhhh MMM hhhh MhMhhh hhhh MM thhhh MMM 2 1 MMM hhh 2 (12) MM hhhh hhhMhM hhhh MM hhhh MMM hhhh MM 3 2 thhh (23) 3 The crossing on the right is s2 = (23). The one on the left if s1 = (12). Reading these from right to left we get: g = (123) = (12)(23) = s1s2. When two crossing lie above each other (so that you don’t know which one is on the left and which is on the right) you can write them in MATH 47A FALL 2008 INTRO TO MATH RESEARCH 5 either order. For example: y (14) x 1 4 1 cFF x FF xx FF xx FF xx FF(12) (12) xx FF xx 2 2 o FF xx 2 FF xx FF xx (23) xxFF xx FF xx FF 3 3 o xx FF 3 xx FF xx(34) (34) FF xx FF xx FF xx FF 4 1 {x 4 This gives: (14) = (12)(34)(23)(12)(34) = s1s3s2s1s3 But the crossings labeled s1 = (12) and s3 = (34) lie above each other. So we can write them in either order: (14) = s3s1s2s3s1 In general, the rule is s s = s s if j i 2 i j j i | − | ≥ Definition 2.3. Suppose that f is a permutation of the letters 1, 2, , n+ 1. Then the length "(f) of f is defined to be the number of pairs· · ·of integers i, j so that (1) 1 i < j n + 1 (2) f(≤i) > f(j≤). In other words, it is the number of pairs of numbers whose order is switched by f. Corollary 2.4. Any permutation f can be written as a product of "(f) simple reflections. Proof. Draw the diagram of the permutation. Move the lines slightly up or down so that there are no triple crossings of lines. Lines i and j will cross if and only if the are in one order on the right (i < j) and in the other order on the left (f(i) > f(j)). Therefore, the number of crossings is equal to "(f) as defined above. The theorem says that each crossing gives one simple reflection and that f is a product of those simple reflections. Therefore, f is a product of "(f) reflections. ! 6 NOTES How large can "(f) be? The maximum possible number occurs when every single pair of in- tegers i < j gets reversed. This is the permutation called w0: 1 2 3 n + 1 w = 0 n + 1 n n 1 · · · 1 ! − · · · " The length of w is the number of all pairs 1 i < j n + 1 which is 0 ≤ ≤ n + 1 n(n + 1) "(w ) = = = 1 + 2 + + n 0 2 2 · · · ! " This is a triangle number. As Tri pointed out in class, w0 can be written as a product of n s ’s, n 1 s ’s, etc. For example, for n = 4, 1 − 2 s4 s3 s3 w0 = s2 s2 s2 s1 s1 s1 s1 = s1s2s1s3s2s4s1s3s2s1 If we want to draw the diagram with 1 at the top and n + 1 = 5 at the bottom, we could also write: s1 s2 s2 w0 = s3 s3 s3 s4 s4 s4 s4 In this method the simple reflection si occurs i times. And this is the method I used on Day 0. There are many different ways to write w0 as n+1 a product of 2 simple reflections. We will study this in detail later. ' ( MATH 47A FALL 2008 INTRO TO MATH RESEARCH 7 3. more group theory Today we talked about the main properties of the elements of a group (although I didn’t define a group yet). I just said that these are all permutations. I changed the format of the class to problem solving, similar to problem sessions, instead of lecturing. 3.1. inverse. Definition 3.1. The inverse of a permutation σ is given by 1 σ− (x) = y where σ(y) = x 1 a) Show that this defines a permutation σ− . b) Find a formula for the inverse of τ = (a1, a2, , ak). 1 1 1 · · · c) Show that (στ)− = τ − σ− . Students found Question (a) confusing so we first did (b) and (c). 3.1.1. inverse of a k-cycle. The inverse of a cycle is given by writing the cycle backwards: 1 τ − = (ak, ak 1, , a2, a1) − · · · This is supposed to obvious, but a proof would go like this: 1 Proof. Let x = ai. Then τ(ai 1) = ai = x.
Load more

Recommended publications
  • 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]
  • 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]
  • 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.
    [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]
  • Chapter 1 GENERAL STRUCTURE and PROPERTIES
    Chapter 1 GENERAL STRUCTURE AND PROPERTIES 1.1 Introduction In this Chapter we would like to introduce the main de¯nitions and describe the main properties of groups, providing examples to illustrate them. The detailed discussion of representations is however demanded to later Chapters, and so is the treatment of Lie groups based on their relation with Lie algebras. We would also like to introduce several explicit groups, or classes of groups, which are often encountered in Physics (and not only). On the one hand, these \applications" should motivate the more abstract study of the general properties of groups; on the other hand, the knowledge of the more important and common explicit instances of groups is essential for developing an e®ective understanding of the subject beyond the purely formal level. 1.2 Some basic de¯nitions In this Section we give some essential de¯nitions, illustrating them with simple examples. 1.2.1 De¯nition of a group A group G is a set equipped with a binary operation , the group product, such that1 ¢ (i) the group product is associative, namely a; b; c G ; a (b c) = (a b) c ; (1.2.1) 8 2 ¢ ¢ ¢ ¢ (ii) there is in G an identity element e: e G such that a e = e a = a a G ; (1.2.2) 9 2 ¢ ¢ 8 2 (iii) each element a admits an inverse, which is usually denoted as a¡1: a G a¡1 G such that a a¡1 = a¡1 a = e : (1.2.3) 8 2 9 2 ¢ ¢ 1 Notice that the axioms (ii) and (iii) above are in fact redundant.
    [Show full text]
  • Spaces of Graphs and Surfaces: on the Work of Søren Galatius
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, January 2012, Pages 73–90 S 0273-0979(2011)01360-X Article electronically published on October 19, 2011 SPACES OF GRAPHS AND SURFACES: ON THE WORK OF SØREN GALATIUS ULRIKE TILLMANN Abstract. We put Soren Galatius’s result on the homology of the automor- phism group of free groups into context. In particular we explain its relation to the Mumford conjecture and the main ideas of the proofs. 1. Introduction Galatius’s most striking result is easy enough to state. Let Σn be the symmetric group on n letters, and let Fn be the free (non-abelian) group with n generators. The symmetric group Σn acts naturally by permutation on the n generators of Fn, and every permutation thus gives rise to an automorphism of the free group. Galatius proves that the map Σn → AutFn in homology induces an isomorphism in degrees less than (n − 1)/2 and in rational homology in degrees less than 2n/3. The homology of the symmetric groups in these ranges is well understood. In particular, in common with all finite groups, it has no non-trivial rational homology. By Galatius’s theorem, in low degrees this is then also true for AutFn: H∗(AutFn) ⊗ Q =0 for 0< ∗ < 2n/3, as had been conjectured by Hatcher and Vogtmann. We will put this result in context and explain the connection with previous work on the mapping class group of surfaces and the homotopy theoretic approach to a conjecture by Mumford in [37] on its rational, stable cohomology.
    [Show full text]
  • Automorphism Groups of Simple Graphs
    AUTOMORPHISM GROUPS OF SIMPLE GRAPHS LUKE RODRIGUEZ Abstract Group and graph theory both provide interesting and meaninful ways of examining relationships between elements of a given set. In this paper we investigate connections between the two. This investigation begins with automorphism groups of common graphs and an introduction of Frucht's Theorem, followed by an in-depth examination of the automorphism groups of generalized Petersen graphs and cubic Hamiltonian graphs in LCF notation. In conclusion, we examine how Frucht's Theorem applies to the specific case of cubic Hamiltonian graphs. 1. Group Theory Fundamentals In the field of Abstract Algebra, one of the fundamental concepts is that of combining particular sets with operations on their elements and studying the resulting behavior. This allows us to consider a set in a more complete way than if we were just considering their elements themselves outside of the context in which they interact. For example, we might be interested in how the set of all integers modulo m behaves under addition, or how the elements of a more abstract set S behave under some set of permutations defined on the elements of S. Both of these are examples of a group. Definition 1.1. A group is a set S together with an operation ◦ such that: (1) For all a; b 2 S, the element a ◦ b = c has the property that c 2 S. (2) For all a; b; c 2 S, the equality a ◦ (b ◦ c) = (a ◦ b) ◦ c holds. (3) There exists an element e 2 S such that a ◦ e = e ◦ a = a for all a 2 S.
    [Show full text]
  • Automorphism Groups of Graphs
    Automorphism groups of graphs ∗ Ashwin Ganesan† Abstract These lecture notes provide an introduction to automorphism groups of graphs. Some special families of graphs are then discussed, espe- cially the families of Cayley graphs generated by transposition sets. Keywords — Automorphism groups of graphs; Cayley graphs; transposition sets. 1 Introduction A permutation of a set Ω is a bijection from Ω to itself. For example, π = (1, 5, 4)(3, 6)(2) = (1, 5, 4)(3, 6) = [5, 2, 6, 1, 4, 3] is a permutation of {1, 2, 3, 4, 5, 6}. We can also write π :1 7→ 5, 2 7→ 2, 3 7→ 6, 4 7→ 1, 5 7→ 4, 6 7→ 3. The inverse of π is π−1 = (1, 4, 5)(3, 6)(2). Sym(Ω) denotes the set of all permutations of Ω. Sn denotes the symmetric group Sym({1,...,n}. The composition of permutations is carried out left to right: (1, 2)(2, 3) = (1, 3, 2). A transposition τ ∈ Sn is a permutation that interchanges two elements and arXiv:1206.6279v1 [cs.DM] 26 Jun 2012 fixes the remaining elements: τ =(i, j). Let Γ := (V, E) be a simple, undirected graph. An automorphism of a graph is a permutation of the vertex set that preserves adjacency. The automorphism group of Γ is the set of permutations of the vertex set that preserve adjacency, i.e., Aut(Γ) := {π ∈ Sym(V ): π(E)= E}. (Some of the ∗Based on an Invited Lecture delivered by the author at the Pre-Conference Workshop on Algebraic Graph theory under the auspices of the 8th Annual Conference of the Academy of Discrete Mathematics and Applications, Virudhunagar, India, June 2012.
    [Show full text]