On a Certain Infinite Permutation Group
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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. -
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. -
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. -
12.6 Further Topics on Simple Groups 387 12.6 Further Topics on Simple Groups
12.6 Further Topics on Simple groups 387 12.6 Further Topics on Simple Groups This Web Section has three parts (a), (b) and (c). Part (a) gives a brief descriptions of the 56 (isomorphism classes of) simple groups of order less than 106, part (b) provides a second proof of the simplicity of the linear groups Ln(q), and part (c) discusses an ingenious method for constructing a version of the Steiner system S(5, 6, 12) from which several versions of S(4, 5, 11), the system for M11, can be computed. 12.6(a) Simple Groups of Order less than 106 The table below and the notes on the following five pages lists the basic facts concerning the non-Abelian simple groups of order less than 106. Further details are given in the Atlas (1985), note that some of the most interesting and important groups, for example the Mathieu group M24, have orders in excess of 108 and in many cases considerably more. Simple Order Prime Schur Outer Min Simple Order Prime Schur Outer Min group factor multi. auto. simple or group factor multi. auto. simple or count group group N-group count group group N-group ? A5 60 4 C2 C2 m-s L2(73) 194472 7 C2 C2 m-s ? 2 A6 360 6 C6 C2 N-g L2(79) 246480 8 C2 C2 N-g A7 2520 7 C6 C2 N-g L2(64) 262080 11 hei C6 N-g ? A8 20160 10 C2 C2 - L2(81) 265680 10 C2 C2 × C4 N-g A9 181440 12 C2 C2 - L2(83) 285852 6 C2 C2 m-s ? L2(4) 60 4 C2 C2 m-s L2(89) 352440 8 C2 C2 N-g ? L2(5) 60 4 C2 C2 m-s L2(97) 456288 9 C2 C2 m-s ? L2(7) 168 5 C2 C2 m-s L2(101) 515100 7 C2 C2 N-g ? 2 L2(9) 360 6 C6 C2 N-g L2(103) 546312 7 C2 C2 m-s L2(8) 504 6 C2 C3 m-s -
Projective Representations of Groups
PROJECTIVE REPRESENTATIONS OF GROUPS EDUARDO MONTEIRO MENDONCA Abstract. We present an introduction to the basic concepts of projective representations of groups and representation groups, and discuss their relations with group cohomology. We conclude the text by discussing the projective representation theory of symmetric groups and its relation to Sergeev and Hecke-Clifford Superalgebras. Contents Introduction1 Acknowledgements2 1. Group cohomology2 1.1. Cohomology groups3 1.2. 2nd-Cohomology group4 2. Projective Representations6 2.1. Projective representation7 2.2. Schur multiplier and cohomology class9 2.3. Equivalent projective representations 10 3. Central Extensions 11 3.1. Central extension of a group 12 3.2. Central extensions and 2nd-cohomology group 13 4. Representation groups 18 4.1. Representation group 18 4.2. Representation groups and projective representations 21 4.3. Perfect groups 27 5. Symmetric group 30 5.1. Representation groups of symmetric groups 30 5.2. Digression on superalgebras 33 5.3. Sergeev and Hecke-Clifford superalgebras 36 References 37 Introduction The theory of group representations emerged as a tool for investigating the structure of a finite group and became one of the central areas of algebra, with important connections to several areas of study such as topology, Lie theory, and mathematical physics. Schur was Date: September 7, 2017. Key words and phrases. projective representation, group, symmetric group, central extension, group cohomology. 2 EDUARDO MONTEIRO MENDONCA the first to realize that, for many of these applications, a new kind of representation had to be introduced, namely, projective representations. The theory of projective representations involves homomorphisms into projective linear groups. Not only do such representations appear naturally in the study of representations of groups, their study showed to be of great importance in the study of quantum mechanics. -
THE CONJUGACY CLASS RANKS of M24 Communicated by Robert Turner Curtis 1. Introduction M24 Is the Largest Mathieu Sporadic Simple
International Journal of Group Theory ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 6 No. 4 (2017), pp. 53-58. ⃝c 2017 University of Isfahan ijgt.ui.ac.ir www.ui.ac.ir THE CONJUGACY CLASS RANKS OF M24 ZWELETHEMBA MPONO Dedicated to Professor Jamshid Moori on the occasion of his seventieth birthday Communicated by Robert Turner Curtis 10 3 Abstract. M24 is the largest Mathieu sporadic simple group of order 244823040 = 2 ·3 ·5·7·11·23 and contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of M24 with respect to the conjugacy classes of all its nonidentity elements. 1. Introduction 10 3 M24 is the largest Mathieu sporadic simple group of order 244823040 = 2 ·3 ·5·7·11·23 and contains all the other Mathieu sporadic simple groups as subgroups. It is a 5-transitive permutation group on a set of 24 points such that: (i) the stabilizer of a point is M23 which is 4-transitive (ii) the stabilizer of two points is M22 which is 3-transitive (iii) the stabilizer of a dodecad is M12 which is 5-transitive (iv) the stabilizer of a dodecad and a point is M11 which is 4-transitive M24 has a trivial Schur multiplier, a trivial outer automorphism group and it is the automorphism group of the Steiner system of type S(5,8,24) which is used to describe the Leech lattice on which M24 acts. M24 has nine conjugacy classes of maximal subgroups which are listed in [4], its ordinary character table is found in [4], its blocks and its Brauer character tables corresponding to the various primes dividing its order are found in [9] and [10] respectively and its complete prime spectrum is given by 2; 3; 5; 7; 11; 23. -
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. -
Representation Growth
Universidad Autónoma de Madrid Tesis Doctoral Representation Growth Autor: Director: Javier García Rodríguez Andrei Jaikin Zapirain Octubre 2016 a Laura. Abstract The main results in this thesis deal with the representation theory of certain classes of groups. More precisely, if rn(Γ) denotes the number of non-isomorphic n-dimensional complex representations of a group Γ, we study the numbers rn(Γ) and the relation of this arithmetic information with structural properties of Γ. In chapter 1 we present the required preliminary theory. In chapter 2 we introduce the Congruence Subgroup Problem for an algebraic group G defined over a global field k. In chapter 3 we consider Γ = G(OS) an arithmetic subgroup of a semisimple algebraic k-group for some global field k with ring of S-integers OS. If the Lie algebra of G is perfect, Lubotzky and Martin showed in [56] that if Γ has the weak Congruence Subgroup Property then Γ has Polynomial Representation Growth, that is, rn(Γ) ≤ p(n) for some polynomial p. By using a different approach, we show that the same holds for any semisimple algebraic group G including those with a non-perfect Lie algebra. In chapter 4 we apply our results on representation growth of groups of the form Γ = D log n G(OS) to show that if Γ has the weak Congruence Subgroup Property then sn(Γ) ≤ n for some constant D, where sn(Γ) denotes the number of subgroups of Γ of index at most n. As before, this extends similar results of Lubotzky [54], Nikolov, Abert, Szegedy [1] and Golsefidy [24] for almost simple groups with perfect Lie algebra to any simple algebraic k-group G. -
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. -
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. -
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 . -
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.