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Author: A. Giambruno
Publisher: American Mathematical Soc.
ISBN: 0821847716
Category : Mathematics
Languages : en
Pages : 283
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Book Description
Represents the proceedings of the conference on Groups, Rings and Group Rings, held July 28 - August 2, 2008, in Ubatuba, Brazil. This title contains results in active research areas in the theory of groups, group rings and algebras (including noncommutative rings), polynomial identities, Lie algebras and superalgebras.
Author: A. Giambruno
Publisher: American Mathematical Soc.
ISBN: 0821847716
Category : Mathematics
Languages : en
Pages : 283
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Book Description
Represents the proceedings of the conference on Groups, Rings and Group Rings, held July 28 - August 2, 2008, in Ubatuba, Brazil. This title contains results in active research areas in the theory of groups, group rings and algebras (including noncommutative rings), polynomial identities, Lie algebras and superalgebras.
Author: Martin J. Taylor
Publisher: Cambridge University Press
ISBN: 0521278708
Category : Mathematics
Languages : en
Pages : 137
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Book Description
This book is a self-contained account of the theory of classgroups of group rings. The guiding philosophy has been to describe all the basic properties of such classgroups in terms of character functions. This point of view is due to A. Frohlich and it achieves a considerable simplification and clarity over previous techniques. A main feature of the book is the introduction of the author's group logarithm, with numerous examples of its application. The main results dealt with are: Ullom's conjecture for Swan modules of p-groups; the self-duality theorem for rings of integers of tame extensions; the fixed-point theorem for determinants of group rings; the existence of Adams operations on classgroups. In addition, the author includes a number of calculations of classgroups of specific families of groups such as generalized dihedral groups, and quaternion and dihedral 2-groups. The work contained in this book should be readily accessible to any graduate student in pure mathematics who has taken a course in the representation theory of finite groups. It will also be of interest to number theorists and algebraic topologists.
Author: K.W. Roggenkamp
Publisher: Birkhäuser
ISBN: 303488611X
Category : Mathematics
Languages : en
Pages : 215
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Book Description
The first part of the book centers around the isomorphism problem for finite groups; i.e. which properties of the finite group G can be determined by the integral group ring ZZG ? The authors have tried to present the results more or less selfcontained and in as much generality as possible concerning the ring of coefficients. In the first section, the class sum correspondence and some related results are derived. This part is the proof of the subgroup rigidity theorem (Scott - Roggenkamp; Weiss) which says that a finite subgroup of the p-adic integral group ring of a finite p-group is conjugate to a subgroup of the finite group. A counterexample to the conjecture of Zassenhaus that group basis are rationally conjugate, is presented in the semilocal situation (Scott - Roggenkamp). To this end, an extended version of Clifford theory for p-adic integral group rings is presented. Moreover, several examples are given to demonstrate the complexity of the isomorphism problem. The second part of the book is concerned with various aspects of the structure of rings of integers as Galois modules. It begins with a brief overview of major results in the area; thereafter the majority of the text focuses on the use of the theory of Hopf algebras. It begins with a thorough and detailed treatment of the required foundational material and concludes with new and interesting applications to cyclotomic theory and to elliptic curves with complex multiplication. Examples are used throughout both for motivation, and also to illustrate new ideas.
Author: Gregory Karpilovsky
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 392
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Book Description
The purpose of this book is to give a self-contained, up-to-date account of the structure of unit groups of classical rings. In so doing, the work draws together four areas of mathematics: ring theory, group theory, group representation theory, and algebraic number theory. The ensuing interplay between these disciplines provides a unique source of enrichment for each of them. The main theme centers on two related problems: to determine the isomorphism class of the unit group (U)R of ring R in terms of natural invariants associated with R; and to find an effective method for the construction of units of ring R. Various threads of the development are tied together to convey a comprehensive picture of the current state of the subject. Examples are provided to help research workers who need to compute explicitly unit groups of certain rings. A familiarity with basic ring-theoretic and group-theoretic concepts is assumed, but a chapter on algebraic preliminaries is included. The text is distinguished by its very clear exposition.
Author: Marlow Anderson
Publisher: CRC Press
ISBN: 1420057111
Category : Mathematics
Languages : en
Pages : 684
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Book Description
Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there
Author: Guo Wenbin
Publisher: Springer Science & Business Media
ISBN: 9401140545
Category : Mathematics
Languages : en
Pages : 270
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Book Description
One of the characteristics of modern algebra is the development of new tools and concepts for exploring classes of algebraic systems, whereas the research on individual algebraic systems (e. g. , groups, rings, Lie algebras, etc. ) continues along traditional lines. The early work on classes of alge bras was concerned with showing that one class X of algebraic systems is actually contained in another class F. Modern research into the theory of classes was initiated in the 1930's by Birkhoff's work [1] on general varieties of algebras, and Neumann's work [1] on varieties of groups. A. I. Mal'cev made fundamental contributions to this modern development. ln his re ports [1, 3] of 1963 and 1966 to The Fourth All-Union Mathematics Con ference and to another international mathematics congress, striking the ories of classes of algebraic systems were presented. These were later included in his book [5]. International interest in the theory of formations of finite groups was aroused, and rapidly heated up, during this time, thanks to the work of Gaschiitz [8] in 1963, and the work of Carter and Hawkes [1] in 1967. The major topics considered were saturated formations, Fitting classes, and Schunck classes. A class of groups is called a formation if it is closed with respect to homomorphic images and subdirect products. A formation is called saturated provided that G E F whenever Gjip(G) E F.
Author: Paolo Aluffi
Publisher: American Mathematical Soc.
ISBN: 147046571X
Category : Education
Languages : en
Pages : 713
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Book Description
Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references.
Author: Donald S. Passman
Publisher: Courier Corporation
ISBN: 0486482065
Category : Mathematics
Languages : en
Pages : 754
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Book Description
"'Highly recommended' by the Bulletin of the London Mathematical Society, this book offers a comprehensive, self-contained treatment of group rings. The subject involves the intersection of two essentially different disciplines, group theory and ring theory. The Bulletin of the American Mathematical Society hailed this treatment as 'a majestic account,' proclaiming it "encyclopedic and lucid." 1985 edition"--
Author: Irving Reiner
Publisher: American Mathematical Soc.
ISBN: 0821816764
Category : Mathematics
Languages : en
Pages : 50
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Book Description
The aim of the lectures is to provide an introduction to recent developments in the theory of class groups and Picard groups. The techniques employed come from the three main areas: algebraic number theory, representation theory of algebras and orders, and algebraic $K$-theory.
Author: Nathan Carter
Publisher: American Mathematical Soc.
ISBN: 1470464330
Category : Education
Languages : en
Pages : 295
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Book Description
Recipient of the Mathematical Association of America's Beckenbach Book Prize in 2012! Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts, but its beauty is lost on students when it is taught in a technical style that is difficult to understand. Visual Group Theory assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective. The more than 300 illustrations in Visual Group Theory bring groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory.
