DOCSLIB.ORG

Computing the Maximal Subsemigroups of a Finite Semigroup

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computing the Maximal Subsemigroups of a Finite Semigroup Computing the maximal subsemigroups of a finite semigroup Wilf Wilson 25th April 2014 1 1 * * 1 2 2 1 * * * * * * 2 * * 2 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 3 3 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 4 4 4 4 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Semigroup diagrams of the four maximal subsemigroups of the partition monoid of degree 3, B3, which do not arise from maximal subgroups of the group of units. See Figure1 and Section 7.3 for more details. For Alan, my little brother. 2 Abstract We characterise the maximal subsemigroups of some special classes of finite semigroup. This then leads us to describe novel algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup. These algorithms have now been implemented in the Semigroups package [8] for GAP [6]. We conclude by detailing the maximal subsemigroups of some specific semigroups, guided by these algorithms. Acknowledgements The original mathematics and implementation of the algorithms discussed in this project was part of a group project. The team consisted, at various points, of Casey Donoven, Robert Hancock, Julius Jonušas, Dr James Mitchell, Dr Yann Péresse, and myself. I enjoyed working with you all. However, the work on this project report is mine alone. I would like to thank Dr Mitchell in particular for his encouraging supervision. 3 Contents 1 Introduction 1 1.1 Intended readership........................................ 1 1.2 Motivation and aims....................................... 1 1.3 Overview of project........................................ 1 1.4 Notation and conventions .................................... 2 1.5 Pre-requisites for understanding................................. 2 2 Graham, Graham and Rhodes6 2.1 A correction............................................ 11 3 Maximal subsemigroups of special classes of finite semigroup 12 3.1 Groups............................................... 12 3.2 Left-zero, right-zero, and other degenerate semigroups .................... 12 3.3 Commutative semigroups .................................... 13 3.4 Commutative bands ....................................... 13 3.5 Monogenic semigroups...................................... 14 3.6 0-simple semigroups ....................................... 15 3.7 Simple semigroups ........................................ 18 3.8 Regular semigroups........................................ 19 4 Description of algorithms for Rees 0-matrix semigroups and Rees matrix semigroups 20 4.1 Algorithm for Rees 0-matrix semigroups............................ 20 4.2 Algorithm for Rees matrix semigroups ............................. 29 5 Description and justification of algorithms for arbitrary semigroups 30 5.1 Compute the maximal subsemigroups which lack a maximal J -class............ 31 5.2 Compute the maximal subsemigroups which lack a non-maximal J -class ......... 33 6 Notes about algorithms 39 6.1 Possible improvements...................................... 39 7 Maximal subsemigroups of specific finite semigroups 40 7.1 Full transformation monoid Tn ................................. 40 7.2 Symmetric inverse monoid In .................................. 41 7.3 Partition monoid Bn ....................................... 41 References 43 Note that many of the diagrams contained in this project convey information through the use of colour. For this reason, it is recommended that this report be read in colour. 4 1 Introduction In this project, we address the question: given a finite semigroup, what are its maximal subsemigroups? Semigroups are renowned for their unrestrictive definition and large number. For example, there are 12,418,001,077,381,302,684 non-equivalent (where equivalence is up to isomorphism or anti-isomorphism) semigroups of order 10 [2]. Only 2 of these are groups. This suggests that answering a question about arbitrary semigroups, especially when considered computationally, can be much more difficult than an- swering an analogous question for groups. Indeed, we will see that in order to answer this question for arbitrary semigroups, we rely on it having already been answered for groups. However, the inherent diversity in semigroup theory does not translate to a complete lack of structure. We find there is much structure present in an arbitrary semigroup, and we will use this to help us answer the question in a practical way. 1.1 Intended readership Many parts of this project are highly technical, and so an understanding of semigroup theory at an advanced undergraduate honours level is required. In particular, we assume that the reader is familiar with all of the topics covered in the module MT5823 Semigroups, which is taught at the University of St Andrews. A familiarity with basic graph theory is also required for several sections of this project. Beyond this level, additional important definitions and results are stated in Section 1.5. We hope that the following pages are both instructive and intriguing to any suitably qualified reader. 1.2 Motivation and aims The maximal subsemigroups of a semigroup are an interesting attribute in their own right. However, being able to know the maximal subsemigroups of a semigroup allows us to answer further questions. For example, it allows us to calculate every subsemigroup of a finite semigroup: we first calculate the maximal subsemigroups, then the maximal subsemigroups of those subsemigroups, and so on, until the process necessarily terminates. Since we are not aware of any previously existing algorithm for maximal subsemigroups, by creating a working algorithm, we are providing a useful tool for other researchers in the field. This is also a demonstration of the power and utility of computational algebra; in particular of the GAP system [6] and the Semigroups package [8] for GAP. We hope the reader will gain a strong understanding of the topic of maximal subsemigroups, and the included algorithms - and perhaps be inspired to explore related topics in computational semigroup theory. 1.3 Overview of project Much of this project is based on the results of Maximal subsemigroups of finite semigroups by Graham, Graham and Rhodes [5]. This paper describes the possible forms of a maximal subsemigroup of a finite semigroup. Section2 is particularly technical, and is entirely devoted entirely to presenting and proving the main results of Graham et al., which we will use throughout the rest of this project. Guided by the results of Section2, we describe the maximal subsemigroups of some special classes of finite semigroup in Section3, such as groups (3.1) and monogenic semigroups (3.5). Next, we describe novel and practical algorithms for calculating maximal subsemigroups, for which we provide pseudocode. Section4 contains algorithms to compute the maximal subsemigroups of a finite regular Rees 0-matrix semigroup over a group, and as a consequence, a finite Rees matrix semigroup over a group. We use these algorithms, along with further results from [5], to describe an algorithm to calculate the maximal subsemigroups of an arbitrary finite semigroup in Section5. We briefly comment on these algorithms in Section6, and suggest future improvements. 1 Finally, in Section7, aided by these algorithms, we calculate the maximal subsemigroups of particularly famous semigroups, such as the full transformation monoid (7.1) and the symmetric inverse monoid (7.2). 1.4 Notation and conventions • S denotes an arbitrary semigroup (S; ∗); we require that S be non-empty. • If the semigroup S is a monoid, then we define S1 to be the semigroup S. If the semigroup S is not a monoid, then we define S1 to be the monoid formed by adjoining the identity 1 to S. • M denotes a maximal subsemigroup of the semigroup S. • For an element a 2 S, Ja denotes the J -class of the element a, i.e. the equivalence class a=J . To S avoid ambiguity when considering multiple semigroups, we write Ja to denote the J -class of the element a in the semigroup S. The other Green’s relations are handled in the same way. • When there is no possibility for confusion, we occasionally omit brackets, for example writing hX; x; yi to denote the semigroup hX [ fx; ygi. • Mappings are composed from left to right. • N denotes the set of natural numbers without zero: f1; 2; 3; :::g. • N0 denotes the set of natural numbers with zero: f0; 1; 2; 3; :::g. 1 2 3 4 5 • When writing transformations we list only the image; for example, the transformation 4 3 5 1 4 is written as (43514). However, when specifically considering a group of permutations, we write in the usual disjoint cycle notation - we shall make clear when this is the case (such as in Example 4.2). 1.5 Pre-requisites for understanding As stated in Section 1.1, it is assumed that the reader is familiar with the basics of semigroup theory, especially with the definitions of subsemigroups, groups, and ideals; with Green’s relations; and with the notion
Load more

Recommended publications
  • ON the SHELLABILITY of the ORDER COMPLEX of the SUBGROUP LATTICE of a FINITE GROUP 1. Introduction We Will Show That the Order C
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 353, Number 7, Pages 2689{2703 S 0002-9947(01)02730-1 Article electronically published on March 12, 2001 ON THE SHELLABILITY OF THE ORDER COMPLEX OF THE SUBGROUP LATTICE OF A FINITE GROUP JOHN SHARESHIAN Abstract. We show that the order complex of the subgroup lattice of a finite group G is nonpure shellable if and only if G is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups. 1. Introduction We will show that the order complex of the subgroup lattice of a finite group G is (nonpure) shellable if and only if G is solvable. The proof of nonshellability in the nonsolvable case involves the determination of the homotopy type of the order complexes of the subgroup lattices of many minimal simple groups. We begin with some history and basic definitions. It is assumed that the reader is familiar with some of the rudiments of algebraic topology and finite group theory. No distinction will be made between an abstract simplicial complex ∆ and an arbitrary geometric realization of ∆. Maximal faces of a simplicial complex ∆ will be called facets of ∆. Definition 1.1. A simplicial complex ∆ is shellable if the facets of ∆ can be ordered σ1;::: ,σn so that for all 1 ≤ i<k≤ n thereexistssome1≤ j<kand x 2 σk such that σi \ σk ⊆ σj \ σk = σk nfxg. The list σ1;::: ,σn is called a shelling of ∆.
    [Show full text]
  • ON the INTERSECTION NUMBER of FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler
    University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-14-2019 ON THE INTERSECTION NUMBER OF FINITE GROUPS Humberto Bautista Serrano University of Texas at Tyler Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Algebra Commons, and the Discrete Mathematics and Combinatorics Commons Recommended Citation Bautista Serrano, Humberto, "ON THE INTERSECTION NUMBER OF FINITE GROUPS" (2019). Math Theses. Paper 9. http://hdl.handle.net/10950/1332 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. ON THE INTERSECTION NUMBER OF FINITE GROUPS by HUMBERTO BAUTISTA SERRANO A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics Kassie Archer, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler April 2019 c Copyright by Humberto Bautista Serrano 2019 All rights reserved Acknowledgments Foremost I would like to express my gratitude to my two excellent advisors, Dr. Kassie Archer at UT Tyler and Dr. Lindsey-Kay Lauderdale at Towson University. This thesis would never have been possible without their support, encouragement, and patience. I will always be thankful to them for introducing me to research in mathematics. I would also like to thank the reviewers, Dr. Scott LaLonde and Dr. David Milan for pointing to several mistakes and omissions and enormously improving the final version of this thesis.
    [Show full text]
  • (Hereditarily) Just Infinite Property in Profinite Groups
    Inverse system characterizations of the (hereditarily) just infinite property in profinite groups Colin D. Reid October 6, 2018 Abstract We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are ‘universal’ in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. This is a corrected and revised version of [8]. 1 Introduction Notation. In this paper, all groups will be profinite groups, all homomorphisms are required to be continuous, and all subgroups are required to be closed; in particular, all references to commutator subgroups are understood to mean the closures of the corresponding abstractly defined subgroups. For an inverse system Λ= {(Gn)n>0, ρn : Gn+1 ։ Gn} of finite groups, we require all the homomorphisms ρn to be surjective. A subscript o will be used to indicate open inclusion, for instance A ≤o B means that A is an open subgroup of B. We use ‘pronilpotent’ and ‘prosoluble’ to mean a group that is the inverse limit of finite nilpotent groups or finite soluble groups respectively, and ‘G-invariant subgroup of H’ to mean a subgroup of H normalized by G. A profinite group G is just infinite if it is infinite, and every nontrivial normal subgroup of G is of finite index; it is hereditarily just infinite if in addition every arXiv:1708.08301v1 [math.GR] 28 Aug 2017 open subgroup of G is just infinite. At first sight the just infinite property is a qualitative one, like that of simplicity: either a group has nontrivial normal subgroups of infinite index, or it does not.
    [Show full text]
  • Arxiv:1509.08090V1 [Math.GR]
    THE CLASS MN OF GROUPS IN WHICH ALL MAXIMAL SUBGROUPS ARE NORMAL AGLAIA MYROPOLSKA Abstract. We investigate the class MN of groups with the property that all maximal subgroups are normal. The class MN appeared in the framework of the study of potential counter-examples to the Andrews-Curtis conjecture. In this note we give various structural properties of groups in MN and present examples of groups in MN and not in MN . 1. Introduction The class MN was introduced in [Myr13] as the class of groups with the property that all maximal subgroups are normal. The study of MN was motivated by the analysis of potential counter-examples to the Andrews-Curtis conjecture [AC65]. It was shown in [Myr13] that a finitely generated group G in the class MN satisfies the so-called “generalised Andrews- Curtis conjecture” (see [BLM05] for the precise definition) and thus cannot confirm potential counter-examples to the original conjecture. Apart from its relation to the Andrews-Curtis conjecutre, the study of the class MN can be interesting on its own. Observe that if a group G belongs to MN then all maximal subgroups of G are of finite index. The latter group property has been considered in the literature for different classes of groups. For instance in the linear setting, Margulis and Soifer [MS81] showed that all maximal subgroups of a finitely generated linear group G are of finite index if and only if G is virtually solvable. The above property also was considered for branch groups, however the results in this direction are partial and far from being as general as for linear groups.
    [Show full text]
  • Math 602 Assignment 1, Fall 2020 Part A
    Math 602 Assignment 1, Fall 2020 Part A. From Gallier{Shatz: Problems 1, 2, 6, 9. Part B. Definition Suppose a group G operates transitively on the left of a set X. (i) We say that the action (G; X) is imprimitive if there exists a non-trivial partition Q of X which is stable under G. Here \non-trivial" means that Q 6= fXg and Q 6= ffxg : x 2 Xg : Such a partition Q will be called a system of imprimitivity for (G; X). (ii) We say that the action (G; X) is primitive if it is not imprimitive. 1. (a) Suppose that (G; X; Q) is a system of imprimitivity for a transitive left action (G; X), Y 2 Q, y 2 Y . Let H = StabG(Y ), K = StabG(y). Prove the following statements. • K < H < G and H operates transitively on Y . • jXj = jY j · jQj, jQj = [G : H], jY j = [H : K]. (These statements hold even if X is infinite.) (b) Suppose that (G; X) is a transitive left action, y 2 X, K := StabG(y), and H is a subgroup of G such that K < H < G. Let Y := H · y ⊂ X, and let Q := fg · Y j g 2 Gg. (Our general notation scheme is that StabG(Y ) := fg 2 G j g · Y = Y g.) Show that (G; X; Q) is a system of imprimitivity. (c) Suppose that (G; X) is a transitive left action, x 2 X. Show that (G; X) is primitive if and only if Gx is a maximal proper subgroup of G.
    [Show full text]
  • Computing the Maximal Subgroups of a Permutation Group I
    Computing the maximal subgroups of a permutation group I Bettina Eick and Alexander Hulpke Abstract. We introduce a new algorithm to compute up to conjugacy the maximal subgroups of a finite permutation group. Or method uses a “hybrid group” approach; that is, we first compute a large solvable normal subgroup of the given permutation group and then use this to split the computation in various parts. 1991 Mathematics Subject Classification: primary 20B40, 20-04, 20E28; secondary 20B15, 68Q40 1. Introduction Apart from being interesting themselves, the maximal subgroups of a group have many applications in computational group theory: They provide a set of proper sub- groups which can be used for inductive calculations; for example, to determine the character table of a group. Moreover, iterative application can be used to investigate parts of the subgroups lattice without the excessive resource requirements of com- puting the full lattice. Furthermore, algorithms to compute the Galois group of a polynomial proceed by descending from the symmetric group via a chain of iterated maximal subgroups, see [Sta73, Hul99b]. In this paper, we present a new approach towards the computation of the conju- gacy classes of maximal subgroups of a finite permutation group. For this purpose we use a “hybrid group” method. This type of approach to computations in permu- tation groups has been used recently for other purposes such as conjugacy classes [CS97, Hul], normal subgroups [Hul98, CS] or the automorphism group [Hol00]. For finite solvable groups there exists an algorithm to compute the maximal sub- groups using a special pc presentation, see [CLG, Eic97, EW].
    [Show full text]
  • Irreducible Character Restrictions to Maximal Subgroups of Low-Rank Classical Groups of Type B and C
    IRREDUCIBLE CHARACTER RESTRICTIONS TO MAXIMAL SUBGROUPS OF LOW-RANK CLASSICAL GROUPS OF TYPE B AND C KEMPTON ALBEE, MIKE BARNES, AARON PARKER, ERIC ROON, AND A.A. SCHAEFFER FRY Abstract Representations are special functions on groups that give us a way to study abstract groups using matrices, which are often easier to understand. In particular, we are often interested in irreducible representations, which can be thought of as the building blocks of all representations. Much of the information about these representations can then be understood by instead looking at the trace of the matrices, which we call the character of the representation. This paper will address restricting characters to subgroups by shrinking the domain of the original representation to just the subgroup. In particular, we will discuss the problem of determining when such restricted characters remain irreducible for certain low-rank classical groups. 1. Introduction Given a finite group G, a (complex) representation of G is a homomorphism Ψ: G ! GLn(C). By summing the diagonal entries of the images Ψ(g) for g 2 G (that is, taking the trace of the matrices), we obtain the corresponding character, χ = Tr◦Ψ of G. The degree of the representation Ψ or character χ is n = χ(1). It is well-known that any character of G can be written as a sum of so- called irreducible characters of G. In this sense, irreducible characters are of particular importance in representation theory, and we write Irr(G) to denote the set of irreducible characters of G. Given a subgroup H of G, we may view Ψ as a representation of H as well, simply by restricting the domain.
    [Show full text]
  • MAT 511 Notes on Maximal Ideals and Subgroups 9/10/13 a Subgroup
    MAT 511 Notes on maximal ideals and subgroups 9/10/13 A subgroup H of a group G is maximal if H 6= G, and, if K is a subgroup of G satisfying H ⊆ K $ G, then H = K. An ideal I of a ring R is maximal if I 6= R, and, if J is an ideal of R satisfying I ⊆ J $ R, then I = J. Zorn's Lemma Suppose P is a nonempty partially-ordered set with the property that every chain in P has an upper bound in P. Then P contains a maximal element. Here a chain C ⊆ P is a totally-ordered subset: for all x; y 2 C, x ≤ y or y ≤ x.A maximal element of P is an element x 2 P with the property x ≤ y =) x = y for all y 2 P. Partially-ordered sets can have several different maximal elements. Zorn's Lemma says, intuitively, if one cannot construct an ever-increasing sequence in P whose terms get arbitrarily large, then there must be a maximal element in P. It is equivalent to the Axiom of Choice: in essence we assume that Zorn's Lemma is true, when we adopt the ZFC axioms as the foundation of mathematics. Theorem 1. If I0 is an ideal of a ring R, and I0 6= R, then there exists a maximal ideal I of R with I0 ⊆ I. Proof. Let P be the set of proper (i.e., 6= R) ideals of R containing I0, ordered by inclusion. Then S P is nonempty, since I0 2 P.
    [Show full text]
  • A New Maximal Subgroup of E8 in Characteristic 3
    A New Maximal Subgroup of E8 in Characteristic 3 David A. Craven, David I. Stewart and Adam R. Thomas March 22, 2021 Abstract We prove the existence and uniqueness of a new maximal subgroup of the algebraic group of type E8 in characteristic 3. This has type F4, and was missing from previous lists of maximal subgroups 3 produced by Seitz and Liebeck–Seitz. We also prove a result about the finite group H = D4(2), that if H embeds in E8 (in any characteristic p) and has two composition factors on the adjoint module then p = 3 and H lies in this new maximal F4 subgroup. 1 Introduction The classification of the maximal subgroups of positive dimension of exceptional algebraic groups [13] is a cornerstone of group theory. In the course of understanding subgroups of the finite groups E8(q) in [3], the first author ran into a configuration that should not occur according to the tables in [13]. We elicit a previously undiscovered maximal subgroup of type F4 of the algebraic group E8 over an algebraically closed field of characteristic 3. This discovery corrects the tables in [13], and the original source [17] on which it depends. Theorem 1.1. Let G be a simple algebraic group of type E8 over an algebraically closed field of char- acteristic 3. Then G contains a unique conjugacy class of simple maximal subgroups of type F4. If X is in this class, then the restriction of the adjoint module L(E8) to X is isomorphic to LX(1000) ⊕ LX(0010), where the first factor is the adjoint module for X of dimension 52 and the second is a simple module of dimension 196 for X.
    [Show full text]
  • Maximal Subgroups of Finite Groups
    MAXIMAL SUBGROUPS OF FINITE GROUPS By LINDSEY-KAY LAUDERDALE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2014 c 2014 Lindsey-Kay Lauderdale ⃝ To Mom, Dad and Lisa ACKNOWLEDGMENTS I would like to thank my advisor Dr. Alexandre Turull, who first sparked my interest in group theory. His endless patience and support were invaluable to the completion of this dissertation. I am also grateful for the support, questions and suggestions of my committee members Dr. Kevin Keating, Dr. David Mazyck, Dr. Paul Robinson and Dr. Peter Sin. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 4 LIST OF TABLES ..................................... 6 ABSTRACT ........................................ 7 CHAPTER 1 DISSERTATION OUTLINE ............................. 9 2 A BRIEF BACKGROUND OF FINITE GROUP THEORY ........... 11 2.1 Introduction ................................... 11 2.2 Maximal Subgroups ............................... 14 2.3 Notation and Some Definitions ......................... 17 2.4 On the Number of Maximal Subgroups .................... 18 3 UPPER BOUNDS FOR THE NUMBER OF MAXIMAL SUBGROUPS .... 20 3.1 Introduction ................................... 20 3.2 Upper Bounds Proven by Others ....................... 21 3.3 Preliminary Results ............................... 22 3.4 Proof of Main Theorem ............................. 23 4 LOWER BOUNDS FOR THE NUMBER OF MAXIMAL SUBGROUPS ...
    [Show full text]
  • Computing Maximal Subgroups of Finite Groups
    Computing maximal subgroups of finite groups John J. Cannon and Derek F. Holt Abstract We describe a practical algorithm for computing representatives of the conjugacy classes of maximal subgroups in a finite group, together with details of its implementation for permutation groups in the MAGMA system. We also describe methods for computing complements of normal subgroups and minimal supplements of normal soluble subgroups of finite groups. 1 Introduction The principal aim of this paper is to describe an algorithm for computing represen- tatives of the conjugacy classes of maximal subgroups of a finite group G, together with our implementation of this algorithm for finite permutation groups within the MAGMA system for computational algebra [2]. It is not a complete algorithm for solving this problem, because it depends on the availability of certain information about the finite nonabelian simple groups, including a detailed knowledge of their maximal subgroups. It can only be applied to groups G for which this information is already available or readily computable for each nonabelian composition factor of G. The precise extent to which it is currently applicable will be discussed later in Section 5. We shall call a finite group a TF-group if it has no nontrivial soluble normal subgroup. (This is because such groups have trivial Fitting subgroups.) The socle of a TF-group is a direct product of nonabelian simple groups. In particular, a group G is called almost simple if its socle is a nonabelian simple group S, and in this case G is isomorphic to a subgroup of the automorphism group Aut(S) of S.
    [Show full text]
  • THE MAXIMAL SUBGROUPS of the SYMMETRIC GROUP Lecture
    THE MAXIMAL SUBGROUPS OF THE SYMMETRIC GROUP MARTINO GARONZI Lecture notes of the Minicourse \The maximal subgroups of the symmetric group" given at the \Escola de Algebra"´ in Unicamp (Campinas, S~aoPaulo, Brazil) on December 3-7 2018. Version 1. I will improve the notes and release a more com- plete version in the future. I am grateful for any comment, criticism and correction. Contents 1. Basic notions about group actions 1 2. The symmetric group 2 3. The maximal subgroups of S5 3 4. Imprimitivity blocks 4 5. Maximal imprimitive subgroups 5 6. Intransitive maximal subgroups 7 7. Primitive maximal subgroups 7 7.1. Characteristically simple groups 7 7.2. Primitive groups 8 7.3. Multiple transitivity 10 7.4. Jordan groups 11 7.5. Primitive actions: O'Nan-Scott 13 References 15 1. Basic notions about group actions All group actions we consider are on the right. G acts on X by the rule (x; g) 7! xg if and only if the map γg : X ! X, x 7! xg induces a homomorphism G ! Sym(X), g 7! γg. The kernel of this homomorphism is the kernel of the action and the action is called faithful if it has trivial kernel. The orbits of the action are the sets O(x) = fxg : g 2 Gg for any x 2 X. The action is called transitive if there is some x 2 X such that O(x) = X (i.e. there is only one orbit), in other words for any x; y 2 X there exists g 2 G with xg = y.
    [Show full text]