Lectures on Finitely Generated Solvable Groups
Lectures on Finitely Generated Solvable Groups

Author: Katalin A. Bencsath

Publisher: Springer Science & Business Media

Published: 2012-10-28

Total Pages: 63

ISBN-13: 1461454506

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Lectures on Finitely Generated Solvable Groups are based on the “Topics in Group Theory" course focused on finitely generated solvable groups that was given by Gilbert G. Baumslag at the Graduate School and University Center of the City University of New York. While knowledge about finitely generated nilpotent groups is extensive, much less is known about the more general class of solvable groups containing them. The study of finitely generated solvable groups involves many different threads; therefore these notes contain discussions on HNN extensions; amalgamated and wreath products; and other concepts from combinatorial group theory as well as commutative algebra. Along with Baumslag’s Embedding Theorem for Finitely Generated Metabelian Groups, two theorems of Bieri and Strebel are presented to provide a solid foundation for understanding the fascinating class of finitely generated solvable groups. Examples are also supplied, which help illuminate many of the key concepts contained in the notes. Requiring only a modest initial group theory background from graduate and post-graduate students, these notes provide a field guide to the class of finitely generated solvable groups from a combinatorial group theory perspective.​

Self-Similar Groups
Self-Similar Groups

Author: Volodymyr Nekrashevych

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 248

ISBN-13: 0821838318

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Self-similar groups (groups generated by automata) initially appeared as examples of groups that are easy to define but have exotic properties like nontrivial torsion, intermediate growth, etc. This book studies the self-similarity phenomenon in group theory and shows its intimate relationship with dynamical systems and more classical self-similar structures, such as fractals, Julia sets, and self-affine tilings. This connection is established through the central topics of the book, which are the notions of the iterated monodromy group and limit space. A wide variety of examples and different applications of self-similar groups to dynamical systems and vice versa are discussed. In particular, it is shown that Julia sets can be reconstructed from the respective iterated monodromy groups and that groups with exotic properties can appear not just as isolated examples, but as naturally defined iterated monodromy groups of rational functions. The book offers important, new mathematics that will open new avenues of research in group theory and dynamical systems. It is intended to be accessible to a wide readership of professional mathematicians.

Cohomology of Finite Groups
Cohomology of Finite Groups

Author: Alejandro Adem

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 333

ISBN-13: 3662062828

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The cohomology of groups has, since its beginnings in the 1920s and 1930s, been the stage for significant interaction between algebra and topology and has led to the creation of important new fields in mathematics, like homological algebra and algebraic K-theory. This is the first book to deal comprehensively with the cohomology of finite groups: it introduces the most important and useful algebraic and topological techniques, and describes the interplay of the subject with those of homotopy theory, representation theory and group actions. The combination of theory and examples, together with the techniques for computing the cohomology of important classes of groups including symmetric groups, alternating groups, finite groups of Lie type, and some of the sporadic simple groups, enable readers to acquire an in-depth understanding of group cohomology and its extensive applications.

Abstract Algebra
Abstract Algebra

Author: Celine Carstensen-Opitz

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2019-09-02

Total Pages: 422

ISBN-13: 3110603993

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A new approach to conveying abstract algebra, the area that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras, that is essential to various scientific disciplines such as particle physics and cryptology. It provides a well written account of the theoretical foundations and it also includes a chapter on cryptography. End of chapter problems help readers with accessing the subjects.

How Groups Grow
How Groups Grow

Author: Avinoam Mann

Publisher: Cambridge University Press

Published: 2011-12-15

Total Pages: 211

ISBN-13: 113950567X

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This book introduces the subject of the growth of groups from scratch, starting with basic definitions and culminating in the seminal results of Gromov and Grigorchuk and more. It is valuable reading for researchers from graduate students up who want to be acquainted with contemporary group theory.

Noncommutative Noetherian Rings
Noncommutative Noetherian Rings

Author: John C. McConnell

Publisher: American Mathematical Soc.

Published: 2001

Total Pages: 658

ISBN-13: 0821821695

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This is a reprinted edition of a work that was considered the definitive account in the subject area upon its initial publication by J. Wiley & Sons in 1987. It presents, within a wider context, a comprehensive account of noncommutative Noetherian rings. The author covers the major developments from the 1950s, stemming from Goldie's theorem and onward, including applications to group rings, enveloping algebras of Lie algebras, PI rings, differential operators, and localization theory. The book is not restricted to Noetherian rings, but discusses wider classes of rings where the methods apply more generally. In the current edition, some errors were corrected, a number of arguments have been expanded, and the references were brought up to date. This reprinted edition will continue to be a valuable and stimulating work for readers interested in ring theory and its applications to other areas of mathematics.

Special Functions and Analysis of Differential Equations
Special Functions and Analysis of Differential Equations

Author: Praveen Agarwal

Publisher: CRC Press

Published: 2020-09-08

Total Pages: 371

ISBN-13: 1000078566

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Differential Equations are very important tools in Mathematical Analysis. They are widely found in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and so on. Recently there has been an increasing interest in and widely-extended use of differential equations and systems of fractional order (that is, of arbitrary order) as better models of phenomena in various physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. Now, new unified presentation and extensive development of special functions associated with fractional calculus are necessary tools, being related to the theory of differentiation and integration of arbitrary order (i.e., fractional calculus) and to the fractional order (or multi-order) differential and integral equations. This book provides learners with the opportunity to develop an understanding of advancements of special functions and the skills needed to apply advanced mathematical techniques to solve complex differential equations and Partial Differential Equations (PDEs). Subject matters should be strongly related to special functions involving mathematical analysis and its numerous applications. The main objective of this book is to highlight the importance of fundamental results and techniques of the theory of complex analysis for differential equations and PDEs and emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Specific topics include but are not limited to Partial differential equations Least squares on first-order system Sequence and series in functional analysis Special functions related to fractional (non-integer) order control systems and equations Various special functions related to generalized fractional calculus Operational method in fractional calculus Functional analysis and operator theory Mathematical physics Applications of numerical analysis and applied mathematics Computational mathematics Mathematical modeling This book provides the recent developments in special functions and differential equations and publishes high-quality, peer-reviewed book chapters in the area of nonlinear analysis, ordinary differential equations, partial differential equations, and related applications.

Topics in Groups and Geometry
Topics in Groups and Geometry

Author: Tullio Ceccherini-Silberstein

Publisher: Springer Nature

Published: 2022-01-01

Total Pages: 468

ISBN-13: 3030881091

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This book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today. The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups.