| Preface: the pursuit of symmetries | p. 1 |
| Finite groups: an introduction | p. 4 |
| Group axioms | p. 5 |
| Finite groups of low order | p. 6 |
| Permutations | p. 19 |
| Basic concepts | p. 22 |
| Conjugation | p. 22 |
| Simple groups | p. 25 |
| Sylow's criteria | p. 27 |
| Semi-direct product | p. 28 |
| Young Tableaux | p. 31 |
| Finite groups: representations | p. 33 |
| Introduction | p. 33 |
| Schur's lemmas | p. 35 |
| The A4 character table | p. 41 |
| Kronecker products | p. 44 |
| Real and complex representations | p. 46 |
| Embeddings | p. 48 |
| Zn character table | p. 52 |
| Dn character table | p. 53 |
| Q2n character table | p. 56 |
| Some semi-direct products | p. 58 |
| Induced representations | p. 61 |
| Invariants | p. 64 |
| Coverings | p. 67 |
| Hilbert spaces | p. 69 |
| Finite Hilbert spaces | p. 69 |
| Fermi oscillators | p. 70 |
| Infinite Hilbert spaces | p. 72 |
| SU (2) | p. 78 |
| Introduction | p. 78 |
| Some representations | p. 82 |
| From Lie algebras to Lie groups | p. 86 |
| SU (2) → SU (1, 1) | p. 89 |
| Selected SU (2) applications | p. 93 |
| The isotropic harmonic oscillator | p. 93 |
| The Bohr atom | p. 95 |
| Isotopic spin | p. 99 |
| SU (3) | p. 102 |
| SU (3) algebra | p. 102 |
| -Basis | p. 106 |
| -Basis | p. 107 |
| ′-Basis | p. 108 |
| The triplet representation | p. 110 |
| The Chevalley basis | p. 112 |
| SU (3) in physics | p. 114 |
| The isotropic harmonic oscillator redux | p. 114 |
| The Elliott model | p. 115 |
| The Sakata model | p. 117 |
| The Eightfold Way | p. 118 |
| Classification of compact simple Lie algebras | p. 123 |
| Classification | p. 124 |
| Simple roots | p. 129 |
| Rank-two algebras | p. 131 |
| Dynkin diagrams | p. 134 |
| Orthonormal bases | p. 140 |
| Lie algebras: representation theory | p. 143 |
| Representation basics | p. 143 |
| A3 fundamentals | p. 144 |
| The Weyl group | p. 149 |
| Orthogonal Lie algebras | p. 151 |
| Spinor representations | p. 153 |
| SO (2n) spinors | p. 154 |
| SO (2n + 1) spinors | p. 156 |
| Clifford algebra construction | p. 159 |
| Casimir invariants and Dynkin indices | p. 164 |
| Embeddings | p. 168 |
| Oscillator representations | p. 178 |
| Verma modules | p. 180 |
| Weyl dimension formula | p. 187 |
| Verma basis | p. 188 |
| Finite groups: the road to simplicity | p. 190 |
| Matrices over Galois fields | p. 192 |
| PSL2 (7) | p. 197 |
| A doubly transitive group | p. 198 |
| Chevalley groups | p. 201 |
| A fleeting glimpse at the sporadic groups | p. 205 |
| Beyond Lie algebras | p. 208 |
| Serre presentation | p. 208 |
| Affine Kac-Moody algebras | p. 210 |
| Super algebras | p. 216 |
| The groups of the Standard Model | p. 221 |
| Space-time symmetries | p. 222 |
| The Lorentz and Poincaré groups | p. 223 |
| The conformal group | p. 231 |
| Beyond space-time symmetries | p. 235 |
| Color and the quark model | p. 239 |
| Invariant Lagrangians | p. 240 |
| Non-Abelian gauge theories | p. 243 |
| The Standard Model | p. 244 |
| Grand Unification | p. 246 |
| Possible family symmetries | p. 249 |
| Finite SU (2) and SO (3) subgroups | p. 249 |
| Finite SU (3) subgroups | p. 252 |
| Exceptional structures | p. 254 |
| Hurwitz algebras | p. 254 |
| Matrices over Hurwitz algebras | p. 257 |
| The Magic Square | p. 259 |
| Properties of some finite groups | p. 265 |
| Properties of selected Lie algebras | p. 277 |
| References | p. 307 |
| Index | p. 308 |
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