Tesis sobre el tema "CHARACTER THEORY, FINITE GROUPS"

The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration.

In this thesis, we will explore the structure of Terwilliger algebras over several different types of finite groups. We will begin by discussing what a Schur ring is, as well as providing many different results and examples of them. Following our discussion on Schur rings, we will move onto discussing association schemes as well as their properties. In particular, we will show every Schur ring gives rise to an association scheme. We will then define a Terwilliger algebra for any finite set, as well as discuss basic properties that hold for all Terwilliger algebras. After specializing to the case of Terwilliger algebras resulting from the orbits of a group, we will explore bounds of the dimension of such a Terwilliger algebra. We will also discuss the Wedderburn decomposition of a Terwilliger algebra resulting from the conjugacy classes of a group for any finite abelian group and any dihedral group.

A finite group H is said to fuse to a finite group G if the class algebra of G is isomorphic to an S-ring over H which is a subalgebra of the class algebra of H. We will also say that G fuses from H. In this case, the classes and characters of H can fuse to give the character table of G. We investigate the case where H is the dihedral group. In many cases, G can be completely determined. In general, G can be proven to have many interesting properties. The theory is developed in terms of S-ring of Schur and Wielandt.

Thesis (Ph.D.)--Kent State University, 2008.
Title from PDF t.p. (viewed Sept. 17, 2009). Advisor: Donald White. Keywords: Huppert's Conjecture; character degrees; nonabelian finite simple groups Includes bibliographical references (p. 103-105).

Consider a finite group G and subgroups H;K of G. We say that H and K permute if HK = KH and call H a permutable subgroup if H permutes with every subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are permutable. We can define, for every finite group G, an arithmetic quantity that measures the probability that two subgroups (chosen uniformly at random with replacement) permute and we call this measure the subgroup permutability degree of G. This measure quantifies, among others, how close a finite group is to being quasi-Dedekind, or, equivalently, nilpotent with modular subgroup lattice. The main body of this thesis is concerned with the behaviour of the subgroup permutability degree of the two families of finite simple groups PSL2(2n), and Sz(q). In both cases the subgroups of the two families of simple groups are completely known and we shall use this fact to establish that the subgroup permutability degree in each case vanishes asymptotically as n or q respectively tends to infinity. The final chapter of the thesis deviates from the main line to examine groups, called F-groups, which behave like nilpotent groups with respect to the Frattini subgroup of quotients. Finally, we present in the Appendix joint research on the distribution of the density of maximal order elements in general linear groups and offer code for computations in GAP related to permutability.

In this thesis, we will see a way to use representation theory and the theory of linear algebraic groups to characterize certain family of finite groups. In Chapter 1, we see the history of preceding work. In particular, J. G. Thompson’s classification of minimal finite simple nonsolvable groups and characterization of solvable groups will be given. In Chapter 2, we will describe some background knowledge underlying this project and notation that will be widely used in this thesis. In Chapter 3, the main theorem originally conjectured by Guralnick and Tiep will be stated together with the base theorem which is a reduced version of main theorem to the case where we have a quasisimple group. Main theorem explains a way to characterize the finite groups with a composition factor of order divisible by two distinct primes p and q as the finite groups containing nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In this thesis we more focus on the proof of showing a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In Chapter 4, we will prove a set of lemmas and proposition which will be used as key tools in the proof of the base theorem. In Chapters 5 to 7, we will establish the base theorem in the cases where a quasisimple group G has its simple quotient isomorphic to alternating groups or sporadic groups (Chapter 5), classical groups (Chapter 6), and exceptional groups (Chapter 7). In Chapter 8, we show that any finite group G admitting nontrivial 2-element x, p- element y, q-element z such that xyz = 1 for two distinct odd primes p and q admits a composition factor of order divisible by pq. Also, we show that the question if a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1 can be reduced to the base theorem.

Let k be a field of characteristic p and let V be a k-vector space. In Chapter 2 of this thesis we classify all unipotent groups G ≤ GL(V ) consisting of bireflections for p not equal to 2: we show that unipotent groups consisting of bireflections are either two-row groups, two-column groups, hook groups or one of two types of exceptional group. The well known theorem of Chevalley-Shephard-Todd shows the importance of (pseudo-)reflection groups to invariant theory. Our interest in bireflection groups is motivated by the theorem of Kemper which tells us if G ≤ GL(V ) is a p-group and the invariant ring k[V ] G is Cohen-Macaulay then G is generated by bireflections. We use our classification to investigate which groups consisting of bireflections have Cohen-Macaulay or complete intersection invariant rings. In Chapter 3 we introduce techniques and notation which we use later to find invariant rings of groups by viewing them as subgroups of Nakajima groups. In Chapter 4 we show that for k = Fp there is a family of hook groups, including all non-abelian hook groups, which have complete intersection invariant rings. It is well known that Cohen-Macaulay invariant rings of p-groups in characteristic p are Gorenstein. There has been speculation by experts in the area, that they might in fact be complete intersections. In Chapter 5 we settle this negatively by giving an example of a p-group which has Cohen-Macaulay but non complete intersection invariant ring. To the best of our knowledge this is the first example of that kind. Finally in Chapter 6 we show that when k = F_p both types of exceptional group have complete intersection invariant rings.

This thesis demonstrates an alternative, concise but informative, method for representing group elements, which will prove particularly useful for the sporadic groups. It explains the theory behind symmetric presentations, and describes the algorithm for working with elements represented in this manner.

Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. In my thesis, I extend earlier work by Douglass and Röhrle for Coxeter groups to the case where G is a complex reflection group of type G(r,p,n) in the notation of Shephard and Todd and X is in the lattice of the reflection arrangement of G. The main result characterizes when the restriction mapping is surjective in terms of the exponents of G and C and their reflection arrangements.

Computational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting questions in group theory, in other branches of mathematics, and in other areas of science. This thesis describes an implementation of a proposal for a Soluble Quotient Algorithm, i.e. a description of the algorithms used and a report on the findings of an empirical study of the behaviour of the programs, and gives an account of an application of the programs. The programs were used for the construction of soluble groups with interesting properties, e.g. for the construction of soluble groups of large derived length which seem to be candidates for groups having efficient presentations. New finite soluble groups of derived length six with trivial Schur multiplier and efficient presentations are described. The methods for finding efficient presentations proved to be only practicable for groups of moderate order. Therefore, for a given derived length soluble groups of small order are of interest. The minimal soluble groups of derived length less than or equal to six are classified.

The foundation of the work of this thesis is based around the involutory progenitor and the finite homomorphic images found therein. This process is developed by Robert T. Curtis and he defines it as 2^{*n} :N {pi w | pi in N, w} where 2^{*n} denotes a free product of n copies of the cyclic group of order 2 generated by involutions. We repeat this process with different control groups and a different array of possible relations to discover interesting groups, such as sporadic, linear, or unitary groups, to name a few. Predominantly this work was produced from transitive groups in 6,10,12, and 18 letters. Which led to identify some appealing groups for this project, such as Janko group J1, Symplectic groups S(4,3) and S(6,2), Mathieu group M12 and some linear groups such as PGL2(7) and L2(11) . With this information, we performed double coset enumeration on some of our findings, M12 over L_2(11) and L_2(31) over D15. We will also prove their isomorphism types with the help of the Jordan-Holder theorem, which aids us in defining the make up of the group. Some examples that we will encounter are the extensions of L_2(31)(center) 2 and A5:2^2.

Molti dei problemi ancora aperti nella teoria delle rappresentazioni dei gruppi finiti riguardano la struttura locale-globale dei gruppi. Sia G un gruppo finito, p un primo che ne divide l'ordine e (K,O,F) un sistema p-modulare di spezzamento. Lo studio locale-globale delle rappresentazioni di G cerca gli invarianti di G che possono essere individuati nei suoi sottogruppi locali, i.e, nel normalizzatore N di un p-sottogruppo D di G, e viceversa. Uno strumento chiave in questo contesto è la corrispondenza di Green, che stabilisce una biezione tra gli OG-reticoli indecomponibili che hanno D (o un suo coniugato in G) come vertice e gli ON-reticoli indecomponibili con vertice D. Lo scopo principale della tesi è lo studio dei reticoli con sorgente lineare e il loro rapporto con le rappresentazioni irriducibili di G e N su K e su F. Gli oggetti principali utilizzati per questo fine sono l'anello di Grothendieck L(G) degli OG-reticoli con sorgente lineare e il suo sottoanello dei reticoli con sorgente banale. Il primo capitolo raccoglie le definizioni e i risultati principali della teoria delle rappresentazioni utilizzati nella tesi. Una particolare attenzione è data alle proprietà dei reticoli con sorgente lineare e alla loro individuazione. Nel Capitolo 2 sono costruite le sezioni canoniche del prodotto tensore con K (risp. con F) definito dall'anello degli OG-reticoli con sorgente lineare (risp. con sorgente banale) all'anello delle KG-rappresentazioni (risp. FG-rappresentazioni). Questo risultato è stato ottenuto seguendo due strategie. La prima prevede la costruzione di mappe duali considerando le "species" degli anelli coinvolti. Il punto di forza di questo approccio è il legame con le tavole delle rappresentazioni definite da Benson, d'altra parte però le mappe considerate prendono valori sulla complessificazione degli anelli. La seconda strategia, che risolve questo problema, consiste nell'utilizzare le formule canoniche di induzione introdotte da Boltje. Infine viene dimostrato che queste due strategie portano allo stesso risultato. Il terzo capitolo è diviso in due parti. Nella prima viene formalmente introdotto l'anello dei reticoli essenziali con sorgente lineare, come conseguenza della definizione di opportune forme bilineari. Nella seconda parte viene analizzato il rapporto tra i reticoli con sorgente banale e vertice massimo e le KG-rappresentazioni irriducibili in due casi particolari: gruppi con sottogruppi normali di indice p e gruppi con sottogruppi di Sylow di ordine p. Nell’ultimo capitolo viene indagato il legame tra la congettura di Alperin-McKay e il gruppo di Grothendieck Lmx(B) dei reticoli con sorgente lineare e vertice massimo in un blocco B di OG. Considerando una delle forme bilineari definite nel capitolo 3 e una opportuna sezione della proiezione canonica di L(B) in Lmx(B), è possibile formulare due nuove congetture (1 e 2), che implicano la congettura di Alperin-McKay e una sua riformulazione di Isaacs e Navarro. Inoltre la congettura 1 e la congettura di Alperin-McKay implicano la riformulazione proposta dai matematici sopracitati. Il risultato principale di questo capitolo è la verifica della congettura 1 in alcuni casi non banali. Per esempio per blocchi "slendid equivant" al loro corrispondente di Brauer. Per un un risultato di R.Rouquier questo vale per tutti i blocchi con gruppo di difetto ciclico. In particolare, questo mostra un inedito legame tra la “splendid form” della congettura di Broué e la riformulazione di Isaacs e Navarro della congettura di Alperin-McKay.
Many investigated and interesting problems in the representation theory of finite groups concern the global and local structure of the groups. Let G be a finite group, p a prime which divides the order of G and (K,O,F) a splitting p-modular system. The local-global study of the representations of G looks for the invariants of G that can be seen in its local subgroups, i.e., the normalizer N of a p-subgroup D of G, and vice versa. A very strong tool in this context is the Green correspondence, which establishes a bijection between the indecomposable OG-lattices with D as a vertex and the indecomposable ON lattices with vertex D. The main scope of this thesis is the study of linear source lattices and their connection with the irreducible representations of G and N both over K and F. The main objects involved for this goal is the Grothendieck ring of linear source OG-lattices L(G) with its subring of trivial source lattices. The first chapter is dedicated to the main results of the representation theory used through all the thesis. Special emphasis is laid on linear source lattices and their detection. In Chapter 2 the canonical sections of the surjective maps given by the tensor product with K for linear source lattices and with F for trivial source lattices are constructed. This result has been obtained following two strategies. The first involves the construction of dual maps defined considering the species of the rings. The strength of this approach is its link to the representation tables defined by Benson; but the maps constructed take values on the complexification of the rings. The canonical induction formulas introduced by R. Boltje turn out to be the solution to bypass this problem. The final result of this part is the proof that these two approaches lead to the same maps. Chapter 3 is divided in two parts. Studying a ring of modules a natural question is if it is possible to define a meaningful bilinear form. In this context the ring of essential linear source lattices arises. In the first part of Chapter 3 it is formally introduced and its species are studied. In the last part the link between trivial source lattices with maximal vertex and irreducible characters is analyzed in two particular cases: groups with normal subgroups of index p and groups with Sylow subgroups of order p. In the last chapter a connection between the Alperin-McKay conjecture and the Grothendieck group Lmx(B) of linear source lattices with maximal vertex in a block B of OG is established. Considering a bilinear form defined in Chapter 3 and a section of the canonical projection of L(B) in Lmx(B), it is possible to state two new conjectures (1 and 2). If both of them are affirmative, then they yield the Alperin-McKay conjecture and one of its refinements due to M. Isaacs and G. Navarro. Moreover, Conjecture 1 and Alperin-McKay conjecture imply its refinement stated by the previously mentioned mathematicians. The main result of this chapter is the proof of Conjecture 1 in some non trivial cases. E.g., for a block splendid equivalent to its Brauer correspondent (for some defect group) Conjecture 1 is positively verified. By a result of R. Rouquier this applies to the case of blocks with cyclic defect groups. This result establishes a new connection between the refinement due to Isaacs and Navarro and the "splendid form" of Broué conjecture.

The goal of this thesis is to examine the actions of finite symmetry groups on aperiodic tilings. To an aperiodic tiling with finite local complexity arising from a primitive substitution rule one can associate a metric space, transformation groupoids, and C*-algebras. Finite symmetry groups of the tiling act on each of these objects and we investigate appropriate constructions on each, namely the orbit space, semidirect product groupoids, and crossed product C*-algebras respectively. Of particular interest are the crossed product C*-algebras; we derive important structure results about them and compute their K-theory.

In this thesis we study the invariant rings for the Sylow p-subgroups of the nite classical groups. We have successfully constructed presentations for the invariant rings for the Sylow p-subgroups of the unitary groups GU(3; Fq2) and GU(4; Fq2 ), the symplectic group Sp(4; Fq) and the orthogonal group O+(4; Fq) with q odd. In all cases, we obtained a minimal generating set which is also a SAGBI basis. Moreover, we computed the relations among the generators and showed that the invariant ring for these groups are a complete intersection. This shows that, even though the invariant rings of the Sylow p-subgroups of the general linear group are polynomial, the same is not true for Sylow p-subgroups of general classical groups. We also constructed the generators for the invariant elds for the Sylow p-subgroups of GU(n; Fq2 ), Sp(2n; Fq), O+(2n; Fq), O-(2n + 2; Fq) and O(2n + 1; Fq), for every n and q. This is an important step in order to obtain the generators and relations for the invariant rings of all these groups.

The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.