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Groups and Symmetries

From Finite Groups to Lie Groups

  • Textbook
  • © 2022
  • Latest edition


  • Connects students to many branches of mathematics (algebra, geometry, and analysis)
  • Applies material to chemistry and physics
  • Includes abundant exercises, many with hints or complete solutions

Part of the book series: Universitext (UTX)

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Table of contents (9 chapters)


About this book

Groups and Symmetries: From Finite Groups to Lie Groups presents an introduction to the theory of group representations and its applications in quantum mechanics. Accessible to advanced undergraduates in mathematics and physics as well as beginning graduate students, the text deals with the theory of representations of finite groups, compact groups, linear Lie groups and their Lie algebras, concisely and in one volume. Prerequisites include calculus and linear algebra.

This new edition contains an additional chapter that deals with Clifford algebras, spin groups, and the theory of spinors, as well as new sections entitled “Topics in history” comprising notes on the history of the material treated within each chapter. (Taken together, they constitute an account of the development of the theory of groups from its inception in the 18th century to the mid-20th.)

References for additional resources and further study are provided in each chapter. All chapters end with exercises of varying degree of difficulty, some of which introduce new definitions and results. The text concludes with a collection of problems with complete solutions making it ideal for both course work and independent study.

Key Topics include:

  • Brisk review of the basic definitions of group theory, with examples
  • Representation theory of finite groups: character theory
  • Representations of compact groups using the Haar measure
  • Lie algebras and linear Lie groups
  • Detailed study of SO(3) and SU(2), and their representations
  • Spherical harmonics
  • Representations of SU(3), roots and weights, with quark theory as a consequence of the mathematical properties of this symmetry group
  • Spin groups and spinors


“The book is very clear and easy to read linearly … . The reader can also cement their understanding with exercises at the end of each chapter, and the book concludes with with an epilogue of worked problems which allow a deeper dive in various directions. … the book is also a great starting point for those with some training in mathematics or physics who wish to broaden their horizons.” (Alastair Litterick, zbMATH 1514.20003, 2023)

Authors and Affiliations

  • Paris, France

    Yvette Kosmann-Schwarzbach

About the author

A former student of the École Normale Supérieure in Paris, Yvette Kosmann-Schwarzbach holds a Doctorat d’État in mathematics as well as a degree in physics from the University of Paris. She has been a professor of mathematics at the University of Lille, at Brooklyn College of the City University of New York, and at the École Polytechnique (France). She has organized numerous conferences, and has held visiting positions and lectured on four continents.

The author of the historical study, The Noether Theorems, Invariance and Conservation Laws in the Twentieth Century (Sources and Studies in the History of Mathematics and Physical Sciences), she has published over 90 research articles in differential geometry, algebra and mathematical physics, and has co-edited The Verdier Memorial Conference on Integrable Systems (Progress in Mathematics), Integrability of Nonlinear Systems (Lecture Notes in Physics) and Discrete Integrable Systems (LectureNotes in Physics).

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