Groups and CharactersGroup representation theory is both elegant and practical, with important applications to quantum mechanics, spectroscopy, crystallography, and other fields in the physical sciences. Until now, however, there have been virtually no accessible treatments of group theory that include representations and characters. The classic works in the field require a high level of mathematical sophistication, and other texts omit representations and characters. Groups and Characters offers an easy-to-follow introduction to the theory of groups and of group characters. Designed as a rapid survey of the subject, this unique text emphasizes examples and applications of the theorems, and avoids many of the longer and more difficult proofs. The author presents group theory through the Sylow Theorems and includes the full subgroup structure of A5. Representations and characters are worked out with numerous character tables, along with real and induced characters that lead to the table for S5. The text includes specific sections that provide the mathematical basis for some of the important applications of group theory in spectroscopy and molecular structure. It also offers numerous exercises-some stressing computation of concrete examples, others stressing development of the mathematical theory. Groups and Characters provides the ideal grounding for more advanced studies with the classic texts, and for more broad-based work in abstract algebra. Furthermore, physical scientists-whose experience with groups and characters may not be rigorous-will find Groups and Characters the ideal means for gaining a sense of the mathematics lying behind the techniques used in applications. |
Contents
Preface | 8 |
Groups and Subgroups | 11 |
Exercises | 18 |
11 | 26 |
Exercises | 41 |
Factor Groups | 51 |
1 | 59 |
Exercises | 67 |
Exercises | 150 |
38 | 158 |
Exercises | 163 |
The Burnside Counting Theorem | 165 |
Real Characters | 173 |
Exercises | 182 |
Exercises | 196 |
The Character Table for S5 | 199 |
1 | 69 |
Exercises | 76 |
Regular Representations | 97 |
Irreducible Representations | 105 |
Representations of Abelian Groups | 119 |
Group Characters | 131 |
1 | 143 |
Space Groups and Semidirect Products | 207 |
Proofs of the Sylow Theorems | 225 |
| 231 | |
| 237 | |
| 238 | |
| 239 | |
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Common terms and phrases
abstract group automorphism C₁ character of degree character of G character table characteristic roots classes of G column completes the proof complex numbers Corollary 15.7 cosets cube cyclic group define Definition denote distinct conjugate classes divides G elements of G elements of order equivalent Example Exercise external direct product factor group finite abelian group finite group given group G group of order hence homomorphism identity induced character integer inverse irreducible character irreducible components irreducible representations isomorphic Lemma Let G Let H Maschke's theorem nonabelian group nonsingular matrix normal subgroup notation number of distinct number of points one-dimensional representations orbits order 12 permutation Proposition proved the following r²c regular representation representation of G result rotation semidirect product subgroup H subgroup of G subgroup of order subset Sylow p-subgroup Sylow theorem t₁ vector verify vertices Young diagrams