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Dieses Buch über Permutationsgruppen bietet neben modernen Beweisen klassischer Ergebnisse, die bislang nicht in Buchform erschienen sind, einen Zugang zur Klassifikation der primitiven Gruppen. Symmetriebetrachtungen von geometrischen Objekten spielen in vielen Naturwissenschaften eine bedeutende Rolle und lassen sich mathematisch durch Permutationsgruppen modellieren. Nachdem wir in diesem Buch eine beliebige Permutationsgruppe in ihre primitiven Bestandteile zerlegt haben, beweisen wir den wichtigen Klassifikationssatz von Aschbacher-O'Nan-Scott, wonach jede primitive Gruppe zu genau einer von fünf Familien gehört. Dieses Resultat erlaubt es zum Beispiel die 2-transitiven Gruppen explizit anzugeben, sodass wir uns im Folgenden auf die primitiven Gruppen, die nicht 2-transitiv sind, konzentrieren können. Die hierfür entwickelte Theorie der Subgrade ermöglicht uns als Anwendung einen Spezialfall des Satzes von Feit-Thompson zu beweisen. Neben zahlreichen Informationen über aktuelle Entwicklungen stehen dem Studierenden über 100 Übungsaufgaben mit vollständigen Lösungen zur Selbstkontrolle zur Verfügung. Vorausgesetzt werden lediglich Kenntnisse einer Algebra-Vorlesung, wobei wir die Grundlagen der elementaren Gruppentheorie im ersten Kapitel wiederholen. Abgerundet wird das Werk durch einen Anhang mit alternativen Beweisen und Quellcodes für die Computeralgebrasysteme GAP und MAGMA. Der Autor Dr. Benjamin Sambale, Fachbereich Mathematik, Technische Universität Kaiserslautern.
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Reihentext + Finite Geometries From the reviews: "Such a vast amount of information as this book contains can only be accomplished in 375 pages by a very economical style of writing... it enables one to have a good look at the forest without being too detracted by the individual trees... The author deserves unstinting praise for the skill, energy, and perseverance which he devoted to this work. The finished product confirms what his many earlier contributions to the subject of finite geometry have already indicated, namely, that he is an undisputed leader in his field." Mathematical Reviews "Finite Geometries" is a very important area of finite mathematics characterized by an interplay of combinatorial, geometric, and algebraic ideas, in which research has been very active and intensive in recent years... makes it clear how large is the field covered by the author in his book. The material is selected most thoroughly, and the author made an effort to collect all that seems to be relevant in finite geometries for the time being... Dembowski's work will be a basic reference book of this field, and it will be considered as a base of the future research... Altogether this is a very well-produced monograph." Publicationes Mathematicae Debrecen 10, tom 16.
Geometry. --- Mathematics. --- Group theory. --- Group Theory and Generalizations.
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This book studies maximal solvable subgroups of classical groups over finite fields. It provides the first modern account of Camille Jordan's classical results, and extends them, giving a classification of maximal irreducible solvable subgroups of general linear groups, symplectic groups, and orthogonal groups over arbitrary finite fields. A subgroup of a group G is said to be maximal solvable if it is maximal among the solvable subgroups of G. The history of this notion goes back to Jordan’s Traité (1870), in which he provided a classification of maximal solvable subgroups of symmetric groups. The main difficulty is in the primitive case, which leads to the problem of classifying maximal irreducible solvable subgroups of general linear groups over a field of prime order. One purpose of this monograph is expository: to give a proof of Jordan’s classification in modern terms. More generally, the aim is to generalize these results to classical groups over arbitrary finite fields, and to provide other results of interest related to irreducible solvable matrix groups. The text will be accessible to graduate students and researchers interested in primitive permutation groups, irreducible matrix groups, and related topics in group theory and representation theory. The detailed introduction will appeal to those interested in the historical background of Jordan’s work.
Group theory. --- Group Theory and Generalizations. --- Solvable groups. --- Finite groups.
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This book introduces readers to key concepts in group theory through engaging puzzles. The early sections of the book show how the rules of group theory emerge naturally from solving puzzles. Different classes of groups, such as cyclic, dihedral and permutation groups are introduced, accompanied by numerous puzzles to facilitate the understanding of the underlying group structures. Later chapters explain how further group theory principles can be applied to puzzle-solving. This book is intended as a highly motivating supplementary text for an undergraduate abstract algebra course. It is also ideal for anyone seeking a fun, hands-on approach to learning group theory. Additionally, the book's many puzzles will be enjoyable for readers already familiar with group theory.
Group theory. --- Mathematics. --- Group Theory and Generalizations. --- General Mathematics.
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"For a finite group G, we denote by [omega](G) the number of Aut(G)-orbits on G, and by o(G) the number of distinct element orders in G. In this paper, we are primarily concerned with the two quantities d(G) :[equals] [omega](G) - o(G) and q(G) :[equals] [omega](G)/ o(G), each of which may be viewed as a measure for how far G is from being an AT-group in the sense of Zhang (that is, a group with [omega](G) [equals] o(G)). We show that the index [absolute value]G : Rad(G) of the soluble radical Rad(G) of G can be bounded from above both by a function in d(G) and by a function in q(G) and o(Rad(G)). We also obtain a curious quantitative characterization of the Fischer-Griess Monster group M"--
Finite groups. --- Automorphisms. --- Group theory and generalizations -- Abstract finite groups -- Arithmetic and combinatorial problems. --- Group theory and generalizations -- Abstract finite groups -- Finite simple groups and their classification. --- Group theory and generalizations -- Abstract finite groups -- Automorphisms.
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This volume focuses on group theory and model theory with a particular emphasis on the interplay of the two areas. The survey papers provide an overview of the developments across group, module, and model theory while the research papers present the most recent study in those same areas. With introductory sections that make the topics easily accessible to students, the papers in this volume will appeal to beginning graduate students and experienced researchers alike. As a whole, this book offers a cross-section view of the areas in group, module, and model theory, covering topics such as DP-minimal groups, Abelian groups, countable 1-transitive trees, and module approximations. The papers in this book are the proceedings of the conference “New Pathways between Group Theory and Model Theory,” which took place February 1-4, 2016, in Mülheim an der Ruhr, Germany, in honor of the editors’ colleague Rüdiger Göbel. This publication is dedicated to Professor Göbel, who passed away in 2014. He was one of the leading experts in Abelian group theory. .
Mathematics. --- Group theory. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra
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Providing a nearly complete selection of up-to-date methods and results on block invariants with respect to their defect groups, this book covers the classical theory pioneered by Brauer, the modern theory of fusion systems introduced by Puig, the geometry of numbers developed by Minkowski, the classification of finite simple groups, and various computer assisted methods. In a powerful combination, these tools are applied to solve many special cases of famous open conjectures in the representation theory of finite groups. Most of the material is drawn from peer-reviewed journal articles, but there are also new previously unpublished results. In order to make the text self-contained, detailed proofs are given whenever possible. Several tables add to the text's usefulness as a reference. The book is aimed at experts in group theory or representation theory who may wish to make use of the presented ideas in their research.
Mathematics. --- Group theory. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra
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This textbook offers students with a basic understanding of group theory a preview of several interesting groups they would not typically encounter until later in their academic careers. By presenting these advanced concepts at this stage, they will gain a deeper understanding of the subject and be motivated to explore more of it. Groups covered include Thompson’s groups, self-similar groups, Lamplighter groups, and Baumslag-Solitar groups. Each chapter focuses on one of these groups, and begins by discussing why they are interesting, how they originated, and why they are important mathematically. A collection of specific references for additional reading, topics for further research, and exercises are included at the end of every chapter to encourage students’ continued education. With its accessible presentation and engaging style, A Sampling of Remarkable Groups is suitable for students in upper-level undergraduate or beginning graduate abstract algebra courses. It will also be of interest to researchers in mathematics, computer science, and related fields.
Group theory. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra
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Differential Geometry. --- Global differential geometry. --- Group Theory and Generalizations. --- Mathematics. --- Number Theory. --- Number theory. --- Real Functions.
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The second edition of this defining handbook provides an up-to-date reference on approaches to the principles and practice of negotiation, group decision-making, and collaboration. It includes the origins, development, and prospects of electronic negotiation, as well as on-line or computer-based arbitration. It constitutes a comprehensive guide to how traditional issues in negotiation, such as knowledge, language, strategy, fairness and justice, have been transformed by technology. The growing field of group decision and negotiation is best described as the empirical, formal, computational, and strategic analysis of group decision-making and negotiation, especially from the viewpoints of organizational behaviour, management science and operations research. The topic crosses many traditional disciplinary boundaries. It has connections to business administration and business strategy, management science, systems engineering, computer science, mathematics, law, economics, psychology, and other social sciences. The first edition greatly strengthened this advancing field. This thoroughly revised and considerably enlarged second edition maintains the approach and philosophy, while adding many important and emerging topics, and an entire section on the frameworks that have created the field. It is a comprehensive, accurate, reliable, and readable reference, and is a major reference volume in the field of group decision and negotiation. .
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