This concise, class-tested book was refined over the authors 30 years as instructors at MIT and the University Federal of Minas Gerais (UFMG) in Brazil. The approach centers on the conviction that teaching group theory along with applications helps students to learn, understand and use it for their own needs. Thus, the theoretical background is confined to introductory chapters. Subsequent chapters develop new theory alongside applications so that students can retain new concepts, build on concepts already learned, and see interrelations between topics. Essential problem sets between chapters aid retention of new material and consolidate material learned in previous chapters.

Research in computational group theory, an active subfield of computational algebra, has emphasised three areas: finite permutation groups, finite solvable groups, and finitely presented groups. This book deals with the third of these areas. The author emphasises the connections with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, computational number theory, and computational commutative algebra. The LLL lattice reduction algorithm and various algorithms for Hermite and Smith normal forms from computational number theory are used to study the abelian quotients of a finitely presented group. The work of Baumslag, Cannonito and Miller on computing nonabelian polycyclic quotients is described as a generalisation of Buchberger's Groebner basis methods to right ideals in the integral group ring of a polycyclic group. Researchers in computational group theory, mathematicians interested in finitely presented groups and theoretical computer scientists will find this book useful.

Presenting classical mechanics, quantum mechanics, and statistical mechanics in an almost completely algebraic setting, this monograph introduces mathematicians, physicists, and engineers to the ideas relating classical and quantum mechanics with Lie algebras and Lie groups. The focus lies on discussing structural properties of mechanics rather than computational techniques.

Geometric group theory is a vibrant subject at the heart of modern mathematics. It is currently enjoying a period of rapid growth and great influence marked by a deepening of its fertile interactions with logic, analysis and large-scale geometry, and striking progress has been made on classical problems at the heart of cohomological group theory. This volume provides the reader with a tour through a selection of the most important trends in the field, including limit groups, quasi-isometric rigidity, non-positive curvature in group theory, and L2-methods in geometry, topology and group theory. Major survey articles exploring recent developments in the field are supported by shorter research papers, which are written in a style that readers approaching the field for the first time will find inviting.

In the large and thriving field of compact transformation groups an important role has long been played by cohomological methods. This book aims to give a contemporary account of such methods, in particular the applications of ordinary cohomology theory and rational homotopy theory with principal emphasis on actions of tori and elementary abelian p-groups on finite-dimensional spaces. For example, spectral sequences are not used in Chapter 1, where the approach is by means of cochain complexes; and much of the basic theory of cochain complexes needed for this chapter is outlined in an appendix. For simplicity, emphasis is put on G-CW-complexes; the refinements needed to treat more general finite-dimensional (or finitistic) G-spaces are often discussed separately. Subsequent chapters give systematic treatments of the Localization Theorem, applications of rational homotopy theory, equivariant Tate cohomology and actions on Poincare duality spaces. Many shorter and more specialized topics are included also. Chapter 2 contains a summary of the main definitions and results from Sullivan's version of rational homotopy theory which are used in the book.

In 1970 Bernd Fischer proved his beautiful theorem classifying the almost simple groups generated by 3-transpositions, and in the process discovered three new sporadic groups, now known as the Fischer groups. Since then, the theory of 3-transposition groups has become an important part of finite simple group theory, but Fischer's work has remained unpublished. 3-Transposition Groups contains the first published proof of Fischer's Theorem, written out completely in one place. Fischer's result, while important and deep (covering a number of complex examples), can be understood by any student with some knowledge of elementary group theory and finite geometry. Thus Part I has minimal prerequisites and could be used as a text for an intermediate level graduate course. Parts II and III are aimed at specialists in finite groups and are a step in the author's program to supply a strong foundation for the theory of sporadic groups.

Many areas of mathematics were deeply influenced or even founded by Hermann Weyl, including geometric foundations of manifolds and physics, topological groups, Lie groups and representation theory, harmonic analysis and analytic number theory as well as foundations of mathematics. In this volume, leading experts present his lasting influence on current mathematics, often connecting Weyl's theorems with cutting edge research in dynamical systems, invariant theory, and partial differential equations. In a broad and accessible presentation, survey chapters describe the historical development of each area alongside up-to-the-minute results, focussing on the mathematical roots evident within Weyl's work.

Presenting groups in a formal, abstract algebraic manner is both useful and powerful, yet it avoids a fascinating geometric perspective on group theory - which is also useful and powerful, particularly in the study of infinite groups. This book presents the modern, geometric approach to group theory, in an accessible and engaging approach to the subject. Topics include group actions, the construction of Cayley graphs, and connections to formal language theory and geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group and Thompson's group. Only exposure to undergraduate-level abstract algebra is presumed, and from that base the core techniques and theorems are developed and recent research is explored. Exercises and figures throughout the text encourage the development of geometric intuition. Ideal for advanced undergraduates looking to deepen their understanding of groups, this book will also be of interest to graduate students and researchers as a gentle introduction to geometric group theory.

Presenting groups in a formal, abstract algebraic manner is both useful and powerful, yet it avoids a fascinating geometric perspective on group theory - which is also useful and powerful, particularly in the study of infinite groups. This book presents the modern, geometric approach to group theory, in an accessible and engaging approach to the subject. Topics include group actions, the construction of Cayley graphs, and connections to formal language theory and geometry. Theorems are balanced by specific examples such as Baumslag-Solitar groups, the Lamplighter group and Thompson's group. Only exposure to undergraduate-level abstract algebra is presumed, and from that base the core techniques and theorems are developed and recent research is explored. Exercises and figures throughout the text encourage the development of geometric intuition. Ideal for advanced undergraduates looking to deepen their understanding of groups, this book will also be of interest to graduate students and researchers as a gentle introduction to geometric group theory.

Representation theory and character theory have proved essential in the study of finite simple groups since their early development by Frobenius. The author begins by presenting the foundations of character theory in a style accessible to advanced undergraduates requiring only a basic knowledge of group theory and general algebra. This theme is then expanded in a self-contained account providing an introduction to the application of character theory to the classification of simple groups. The book follows both strands of the theory: the exceptional characters of Suzuki and Feit and the block character theory of Brauer and includes refinements of original proofs that have become available as the subject has grown. This account will be of value as a textbook for students with some background in group theory, and as a reference for specialists and researchers in the field.

This book describes various approaches to the Inverse Galois Problem, a classical unsolved problem of mathematics posed by Hilbert at the beginning of the century. It brings together ideas from group theory, algebraic geometry and number theory, topology, and analysis. Assuming only elementary algebra and complex analysis, the author develops the necessary background from topology, Riemann surface theory and number theory. The first part of the book is quite elementary, and leads up to the basic rigidity criteria for the realization of groups as Galois groups. The second part presents more advanced topics, such as braid group action and moduli spaces for covers of the Riemann sphere, GAR- and GAL- realizations, and patching over complete valued fields. Graduate students and mathematicians from other areas (especially group theory) will find this an excellent introduction to a fascinating field.

This book is the first volume in a two-volume set, which will provide the complete proof of classification of two important classes of geometries, closely related to each other: Petersen and tilde geometries. There is an infinite family of tilde geometries associated with non-split extensions of symplectic groups over a field of two elements. Besides that there are twelve exceptional Petersen and tilde geometries. These exceptional geometries are related to sporadic simple groups, including the famous Monster group and this volume gives a construction for each of the Petersen and tilde geometries which provides an independent existence proof for the corresponding automorphism group. Important applications of Petersen and Tilde geometries are considered, including the so-called Y-presentations for the Monster and related groups, and a complete indentification of Y-groups is given. This is an essential purchase for researchers into finite group theory, finite geometries and algebraic combinatorics.

Based on the third International Conference on Symmetries, Differential Equations and Applications (SDEA-III), this proceedings volume highlights recent important advances and trends in the applications of Lie groups, including a broad area of topics in interdisciplinary studies, ranging from mathematical physics to financial mathematics. The selected and peer-reviewed contributions gathered here cover Lie theory and symmetry methods in differential equations, Lie algebras and Lie pseudogroups, super-symmetry and super-integrability, representation theory of Lie algebras, classification problems, conservation laws, and geometrical methods. The SDEA III, held in honour of the Centenary of Noether's Theorem, proven by the prominent German mathematician Emmy Noether, at Istanbul Technical University in August 2017 provided a productive forum for academic researchers, both junior and senior, and students to discuss and share the latest developments in the theory and applications of Lie symmetry groups. This work has an interdisciplinary appeal and will be a valuable read for researchers in mathematics, mechanics, physics, engineering, medicine and finance.

Sporadic Groups is the first step in a programme to provide a uniform, self-contained treatment of the foundational material on the sporadic finite simple groups. The classification of the finite simple groups is one of the premier achievements of modern mathematics. The classification demonstrates that each finite simple group is either a finite analogue of a simple Lie group or one of 26 pathological sporadic groups. Sporadic Groups provides for the first time a self-contained treatment of the foundations of the theory of sporadic groups accessible to mathematicians with a basic background in finite groups such as in the author's text Finite Group Theory. Introductory material useful for studying the sporadics, such as a discussion of large extraspecial 2-subgroups and Tits' coset geometries, opens the book. A construction of the Mathieu groups as the automorphism groups of Steiner systems follows. The Golay and Todd modules, and the 2-local geometry for M24 are discussed. This is followed by the standard construction of Conway of the Leech lattice and the Conway group. The Monster is constructed as the automorphism group of the Griess algebra using some of the best features of the approaches of Griess, Conway, and Tits, plus a few new wrinkles. Researchers in finite group theory will find this text invaluable. The subjects treated will interest combinatorists, number theorists, and conformal field theorists.

Line up a deck of 52 cards on a table. Randomly choose two cards and switch them. How many switches are needed in order to mix up the deck? Starting from a few concrete problems such as random walks on the discrete circle and the finite ultrametric space this book develops the necessary tools for the asymptotic analysis of these processes. This detailed study culminates with the case-by-case analysis of the cut-off phenomenon discovered by Persi Diaconis. This self-contained text is ideal for graduate students and researchers working in the areas of representation theory, group theory, harmonic analysis and Markov chains. Its topics range from the basic theory needed for students new to this area, to advanced topics such as the theory of Green's algebras, the complete analysis of the random matchings, and the representation theory of the symmetric group.

How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive and up-to-date account of research into this question spanning almost fifty years. The authors presuppose an undergraduate knowledge of group theory, up to and including Sylow's Theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory, and a very little cohomology theory - but most of the basics are expounded here and the book is more or less self-contained. Although it is principally devoted to a connected exposition of an agreeable theory, the book does also contain some material that has not hitherto been published. It is designed to be used as a graduate text but also as a handbook for established research workers in group theory.

The decomposition of the space L2 (G(Q)\G(/A)), where G is a reductive group defined over (Q and /A is the ring of adeles of (Q, is a deep problem at the intersection of number and group theory. Langlands reduced this decomposition to that of the (smaller) spaces of cuspidal automorphic forms for certain subgroups of G. The present book describes this proof in detail. The starting point is the theory of automorphic forms, which can also serve as a first step towards understanding the Arthur-Selberg trace formula. To make the book reasonably self-contained, the authors have also provided essential background to subjects such as automorphic forms, Eisenstein series, Eisenstein pseudo-series (or wave-packets) and their properties. It is thus also an introduction, suitable for graduate students, to the theory of automorphic forms, written using contemporary terminology. It will be welcomed by number theorists, representation theorists, and all whose work involves the Langlands program.

This thoroughly revised and updated version of the popular textbook on abstract algebra introduces students to easily understood problems and concepts. John Humphreys and Mike Prest include many examples and exercises throughout the book to make it more appealing to students and instructors. The second edition features new sections on mathematical reasoning and polynomials. In addition, three chapters have been completely rewritten and all others have been updated. First Edition Pb (1990): 0-521-35938-4

In 1999 a number of eminent mathematicians were invited to Bielefeld to present lectures at a conference on topological, combinatorial and arithmetic aspects of (infinite) groups. The present volume consists of survey and research articles invited from participants in this conference. Topics covered include topological finiteness properties of groups, Kac-Moody groups, the theory of Euler characteristics, the connection between groups, formal languages and automata, the Magnus-Nielsen method for one-relator groups, atomic and just infinite groups, topology in permutation groups, probabilistic group theory, the theory of subgroup growth, hyperbolic lattices in dimension three, generalised triangle groups and reduction theory. All contributions are written in a relaxed and attractive style, accessible not only to specialists, but also to good graduate and post-graduate students, who will find inspiration for a number of basic research projects at various levels of technical difficulty.

Can a sense of belonging increase life satisfaction? Why do we sometimes feel lonely? How can we sustain lasting human connections? The Psychology of Belonging explores why feeling like we belong is so important throughout our lives, from childhood to old age, irrespective of culture, race or geography. With its virtues and shortcomings, belonging to groups such as families, social groups, schools, workplaces and communities is fundamental to our identity and wellbeing, even in a time when technology has changed the way we connect with each other. In a world where loneliness and social isolation is on the rise, The Psychology of Belonging shows how meaningful connections can build a sense of belonging for all of us.

This two-volume set contains selected papers from the conference Groups St. Andrews 2001 in Oxford. Contributed by leading researchers, the articles cover a wide spectrum of modern group theory. Contributions based on lecture courses given by five main speakers are included with refereed survey and research articles.

This two-volume set contains selected papers from the conference Groups St. Andrews 2001 in Oxford. Contributed by leading researchers, the articles cover a wide spectrum of modern group theory. Contributions based on lecture courses given by five main speakers are included with refereed survey and research articles. The Groups St. Andrews proceedings volumes represent a view of the state of the art in group theory and often play an important role in future developments in the subject.

At the crossroads of representation theory, algebraic geometry and finite group theory, this 2004 book blends together many of the main concerns of modern algebra, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of etale cohomology, leading to the proof of the recent Bonnafe-Rouquier theorems. The second is a straightforward and simplified account of the Dipper-James theorems relating irreducible characters and modular representations. The final theme is local representation theory. One of the main results here is the authors' version of Fong-Srinivasan theorems. Throughout the text is illustrated by many examples and background is provided by several introductory chapters on basic results and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.

The first edition of this book was the indispensable reference for researchers in the theory of pro-p groups. In this second edition the presentation has been improved and important new material has been added. The first part of the book is group-theoretic. It develops the theory of pro-p groups of finite rank, starting from first principles and using elementary methods. Part II introduces p-adic analytic groups: by taking advantage of the theory developed in Part I, it is possible to define these, and derive all the main results of p-adic Lie theory, without having to develop any sophisticated analytic machinery. Part III, consisting of new material, takes the theory further. Among those topics discussed are the theory of pro-p groups of finite coclass, the dimension subgroup series, and its associated graded Lie algebra. The final chapter sketches a theory of analytic groups over pro-p rings other than the p-adic integers.