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This landmark among mathematics texts applies group theory to quantum mechanics, first covering unitary geometry, quantum theory, groups and their representations, then applications themselves — rotation, Lorentz, permutation groups, symmetric permutation groups, and the algebra of symmetric transformations.
Group theory represents one of the most fundamental elements of mathematics. Indispensable in nearly every branch of the field, concepts from the theory of groups also have important applications beyond mathematics, in such areas as quantum mechanics and crystallography. Hans J. Zassenhaus, a pioneer in the study of group theory, has designed this useful, well-written, graduate-level text to acquaint the reader with group-theoretic methods and to demonstrate their usefulness as tools in the solution of mathematical and physical problems. Starting with an exposition of the fundamental concepts of group theory, including an investigation of axioms, the calculus of complexes, and a theorem of Frobenius, the author moves on to a detailed investigation of the concept of homomorphic mapping, along with an examination of the structure and construction of composite groups from simple components. The elements of the theory of p-groups receive a coherent treatment, and the volume concludes with an explanation of a method by which solvable factor groups may be split off from a finite group. Many of the proofs in the text are shorter and more transparent than the usual, older ones, and a series of helpful appendixes presents material new to this edition. This material includes an account of the connections between lattice theory and group theory, and many advanced exercises illustrating both lattice-theoretical ideas and the extension of group-theoretical concepts to multiplicative domains.
265 challenging problems in all phases of group theory, gathered for the most part from papers published since 1950, although some classics are included.
Here is a clear, well-organized coverage of the most standard theorems, including isomorphism theorems, transformations and subgroups, direct sums, abelian groups, and more. This undergraduate-level text features more than 500 exercises.
" A group is defined by means of the laws of combinations of its symbols," according to a celebrated dictum of Cayley. And this is probably still as good a one-line explanation as any. The concept of a group is surely one of the central ideas of mathematics. Certainly there are a few branches of that science in which groups are not employed implicitly or explicitly. Nor is the use of groups confined to pure mathematics. Quantum theory, molecular and atomic structure, and crystallography are just a few of the areas of science in which the idea of a group as a measure of symmetry has played an important part. The theory of groups is the oldest branch of modern algebra. Its origins are to be found in the work of Joseph Louis Lagrange (1736-1813), Paulo Ruffini (1765-1822), and Evariste Galois (1811-1832) on the theory of algebraic equations. Their groups consisted of permutations of the variables or of the roots of polynomials, and indeed for much of the nineteenth century all groups were finite permutation groups. Nevertheless many of the fundamental ideas of group theory were introduced by these early workers and their successors, Augustin Louis Cauchy (1789-1857), Ludwig Sylow (1832-1918), Camille Jordan (1838-1922) among others. The concept of an abstract group is clearly recognizable in the work of Arthur Cayley (1821-1895) but it did not really win widespread acceptance until Walther von Dyck (1856-1934) introduced presentations of groups.
"The book is a pleasure to read. There is no question but that it will become, and deserves to be, a widely used textbook and reference." — Bulletin of the American Mathematical Society. Character theory provides a powerful tool for proving theorems about finite groups. In addition to dealing with techniques for applying characters to "pure" group theory, a large part of this book is devoted to the properties of the characters themselves and how these properties reflect and are reflected in the structure of the group. Chapter I consists of ring theoretic preliminaries. Chapters 2 to 6 and 8 contain the basic material of character theory, while Chapter 7 treats an important technique for the application of characters to group theory. Chapter 9 considers irreducible representations over arbitrary fields, leading to a focus on subfields of the complex numbers in Chapter 10. In Chapter 15 the author introduces Brauer’s theory of blocks and "modular characters." Remaining chapters deal with more specialized topics, such as the connections between the set of degrees of the irreducible characters and structure of a group. Following each chapter is a selection of carefully thought out problems, including exercises, examples, further results and extensions and variations of theorems in the text. Prerequisites for this book are some basic finite group theory: the Sylow theorems, elementary properties of permutation groups and solvable and nilpotent groups. Also useful would be some familiarity with rings and Galois theory. In short, the contents of a first-year graduate algebra course should be sufficient preparation.

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