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Groups Modular Mathematics Series

Author: Camilla Jordan
Publisher: Butterworth-Heinemann
ISBN: 0080571654
Size: 22.10 MB
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This text provides an introduction to group theory with an emphasis on clear examples. The authors present groups as naturally occurring structures arising from symmetry in geometrical figures and other mathematical objects. Written in a 'user-friendly' style, where new ideas are always motivated before being fully introduced, the text will help readers to gain confidence and skill in handling group theory notation before progressing on to applying it in complex situations. An ideal companion to any first or second year course on the topic.

Groups St Andrews 1989

Author: Groups-St Andrews (1989 : St Andrews)
Publisher: Cambridge University Press
ISBN: 9780521406697
Size: 11.92 MB
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These two volumes contain selected papers presented at the international conference on group theory held at St. Andrews in 1989. The themes of the conference were combinatorial and computational group theory; leading group theorists, including J.A. Green, N.D. Gupta, O.H. Kegel and J.G. Thompson, gave courses whose content is reproduced here. Also included are refereed papers presented at the meeting.

Arithmetic Fundamental Groups And Noncommutative Algebra

Author: Karen Redrobe Beckman
Publisher: American Mathematical Soc.
ISBN: 0821820362
Size: 26.42 MB
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The arithmetic and geometry of moduli spaces and their fundamental groups are a very active research area. This book offers a complete overview of developments made over the last decade. The papers in this volume examine the geometry of moduli spaces of curves with a function on them. The main players in Part 1 are the absolute Galois group $G_{\mathbb Q}$ of the algebraic numbers and its close relatives. By analyzing how $G_{\mathbb Q}$ acts on fundamental groups defined by Hurwitz moduli problems, the authors achieve a grand generalization of Serre's program from the 1960s.Papers in Part 2 apply $\theta$-functions and configuration spaces to the study of fundamental groups over positive characteristic fields. In this section, several authors use Grothendieck's famous lifting results to give extensions to wildly ramified covers. Properties of the fundamental groups have brought collaborations between geometers and group theorists. Several Part 3 papers investigate new versions of the genus 0 problem. In particular, this includes results severely limiting possible monodromy groups of sphere covers. Finally, Part 4 papers treat Deligne's theory of Tannakian categories and arithmetic versions of the Kodaira-Spencer map. This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry.

Discrete Groups And Geometry

Author: Conference on discrete groups and geometry
Publisher: Cambridge University Press
ISBN: 9780521429320
Size: 28.34 MB
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This volume contains a selection of refereed papers presented in honour of A.M. Macbeath, one of the leading researchers in the area of discrete groups. The subject has been of much current interest of late as it involves the interaction of a number of diverse topics such as group theory, hyperbolic geometry, and complex analysis.

Theta Constants Riemann Surfaces And The Modular Group

Author: Hershel M. Farkas
Publisher: American Mathematical Soc.
ISBN: 0821813927
Size: 13.42 MB
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There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova. Theta functions are also classically connected with Riemann surfaces and with the modular group $\Gamma = \mathrm{PSL}(2,\mathbb{Z})$, which provide another path for insights into number theory. Farkas and Kra, well-known masters of the theory of Riemann surfaces and the analysis of theta functions, uncover here interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. For instance, the authors use this approach to derive congruences discovered by Ramanujan for the partition function, with the main ingredient being the construction of the same function in more than one way. The authors also obtain a variant on Jacobi's famous result on the number of ways that an integer can be represented as a sum of four squares, replacing the squares by triangular numbers and, in the process, obtaining a cleaner result. The recent trend of applying the ideas and methods of algebraic geometry to the study of theta functions and number theory has resulted in great advances in the area. However, the authors choose to stay with the classical point of view. As a result, their statements and proofs are very concrete. In this book the mathematician familiar with the algebraic geometry approach to theta functions and number theory will find many interesting ideas as well as detailed explanations and derivations of new and old results. Highlights of the book include systematic studies of theta constant identities, uniformizations of surfaces represented by subgroups of the modular group, partition identities, and Fourier coefficients of automorphic functions. Prerequisites are a solid understanding of complex analysis, some familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and an interest in number theory. The book contains summaries of some of the required material, particularly for theta functions and theta constants. Readers will find here a careful exposition of a classical point of view of analysis and number theory. Presented are numerous examples plus suggestions for research-level problems. The text is suitable for a graduate course or for independent reading.

Proceedings Of Groups St Andrews 1985

Author: E. F. Robertson
Publisher: Cambridge University Press
ISBN: 9780521338547
Size: 80.95 MB
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A current picture of progress and research in group theory is provided by five leading group theorists Bachmuth, Baumslag, Neumann, Roseblade and Tits.

Modular Representation Theory Of Finite Groups

Author: Michael J. Collins
Publisher: Walter de Gruyter
ISBN: 3110889161
Size: 13.45 MB
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This book is an outgrowth of a Research Symposium on the Modular Representation Theory of Finite Groups, held at the University of Virginia in May 1998. The main themes of this symposium were representations of groups of Lie type in nondefining (or cross) characteristic, and recent developments in block theory. Series of lectures were given by M. Geck, A. Kleshchev and R. Rouquier, and their brief was to present material at the leading edge of research but accessible to graduate students working in the field. The first three articles are substantial expansions of their lectures, and each provides a complete account of a significant area of the subject together with an extensive bibliography. The remaining articles are based on some of the other lectures given at the symposium; some again are full surveys of the topic covered while others are short, but complete, research articles. The opportunity has been taken to produce a book of enduring value so that this is not a conference proceedings in the conventional sense. Material has been updated so that this book, through its own content and in its extensive bibliographies, will serve as an invaluable resource for all those working in the area, whether established researchers or graduate students who wish to gain a general knowledge of the subject starting from a single source.

The Theory Of Groups

Author: Aleksandr Gennadievich Kurosh
Publisher: American Mathematical Soc.
ISBN: 0821834770
Size: 54.73 MB
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This book is translated from the second Russian edition and with added notes by K.A. Hirsch. ""Teoriya Grupp"" by Kurosh was widely acclaimed, in its first edition, as the first modern text on the general theory of groups, with the major emphasis on infinite groups. The decade that followed brought about a remarkable growth and maturity in the theory of groups, so that this second edition, in English translation, represents a complete rewriting of the first edition.The book can be used as a beginning text, the only requirement being some mathematical maturity and a knowledge of the elements of transfinite numbers. Many new sections were added to this second edition, and many old ones were completely revised: the theory of abelian groups was significantly revised; many significant additions were made to the section on the theory of free groups and free products; an entire chapter is devoted to group extensions; and the deep changes in the theory of solvable and nilpotent groups - one of the large and rich branches of the theory of groups - are covered in this work. Each volume concludes with Editor's Notes and a Bibliography.