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§ 4 The group algebra

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Character Theory of Finite Groups
This chapter is in the book Character Theory of Finite Groups
© 2011 Walter de Gruyter GmbH & Co. KG, Genthiner Str. 13, 10785 Berlin.

© 2011 Walter de Gruyter GmbH & Co. KG, Genthiner Str. 13, 10785 Berlin.

Chapters in this book

  1. Frontmatter I
  2. Introduction 1
  3. Examples of groups 3
  4. Notations and results from group theory 5
  5. § 1 Representations and representation modules 10
  6. § 2 Simple and semisimple modules 17
  7. § 3 Orthogonality relations 28
  8. § 4 The group algebra 46
  9. § 5 Characters of abelian groups 58
  10. § 6 Degrees of irreducible representations 66
  11. § 7 Characters of some small groups 79
  12. § 8 Products of representations and characters 96
  13. § 9 On the number of solutions of gm = 1 in a group 108
  14. § 10 A theorem of A. Hurwitz on multiplicative sums of squares 124
  15. § 11 Permutation representations and characters 131
  16. § 12 The class number 163
  17. § 13 Real characters and real representations 171
  18. § 14 Coprime action 184
  19. § 15 Groups of order paqb 190
  20. § 16 Frobenius groups 196
  21. § 17 Induced characters 215
  22. § 18 Brauer’s permutation lemma and Glauberman’s character correspondence 233
  23. § 19 Clifford Theory 1 252
  24. § 20 Projective representations 273
  25. § 21 Clifford Theory 2 285
  26. § 22 Extensions of characters 294
  27. § 23 Degree pattern and group structure 310
  28. § 24 Monomial groups 318
  29. § 25 Representations of wreath products 338
  30. § 26 Characters of p-groups 351
  31. § 27 Groups with a small number of character degrees 375
  32. § 28 Linear groups 394
  33. § 29 Finite subgroups of SO(3, ℝ) and SU(2, ℂ) 402
  34. § 30 The degree graph 413
  35. § 31 Groups all of whose character degrees are primes 416
  36. § 32 Two special degree problems 432
  37. § 33 Lengths of conjugacy classes 443
  38. § 34 R. Brauer’s theorems on the character ring 466
  39. § 35 Applications of Brauer’s theorems 477
  40. § 36 Artin’s induction theorem 489
  41. § 37 Splitting fields 502
  42. § 38 The Schur index 507
  43. § 39 Integral representations 526
  44. § 40 Isaacs π-special characters 538
  45. § 41 Three arithmetical applications 548
  46. § 42 Small kernels and faithful irreducible characters 554
  47. § 43 TI-sets 567
  48. § 44 Involutions 573
  49. § 45 Groups whose Sylow-2-subgroups are generalized quaternion groups 580
  50. § 46 Perfect Frobenius complements 597
  51. Books 606
  52. Bibliography 607
  53. Index of names 614
  54. Index of subjects 617
https://doi.org/10.1515/9783110809237.46

Chapters in this book

  1. Frontmatter I
  2. Introduction 1
  3. Examples of groups 3
  4. Notations and results from group theory 5
  5. § 1 Representations and representation modules 10
  6. § 2 Simple and semisimple modules 17
  7. § 3 Orthogonality relations 28
  8. § 4 The group algebra 46
  9. § 5 Characters of abelian groups 58
  10. § 6 Degrees of irreducible representations 66
  11. § 7 Characters of some small groups 79
  12. § 8 Products of representations and characters 96
  13. § 9 On the number of solutions of gm = 1 in a group 108
  14. § 10 A theorem of A. Hurwitz on multiplicative sums of squares 124
  15. § 11 Permutation representations and characters 131
  16. § 12 The class number 163
  17. § 13 Real characters and real representations 171
  18. § 14 Coprime action 184
  19. § 15 Groups of order paqb 190
  20. § 16 Frobenius groups 196
  21. § 17 Induced characters 215
  22. § 18 Brauer’s permutation lemma and Glauberman’s character correspondence 233
  23. § 19 Clifford Theory 1 252
  24. § 20 Projective representations 273
  25. § 21 Clifford Theory 2 285
  26. § 22 Extensions of characters 294
  27. § 23 Degree pattern and group structure 310
  28. § 24 Monomial groups 318
  29. § 25 Representations of wreath products 338
  30. § 26 Characters of p-groups 351
  31. § 27 Groups with a small number of character degrees 375
  32. § 28 Linear groups 394
  33. § 29 Finite subgroups of SO(3, ℝ) and SU(2, ℂ) 402
  34. § 30 The degree graph 413
  35. § 31 Groups all of whose character degrees are primes 416
  36. § 32 Two special degree problems 432
  37. § 33 Lengths of conjugacy classes 443
  38. § 34 R. Brauer’s theorems on the character ring 466
  39. § 35 Applications of Brauer’s theorems 477
  40. § 36 Artin’s induction theorem 489
  41. § 37 Splitting fields 502
  42. § 38 The Schur index 507
  43. § 39 Integral representations 526
  44. § 40 Isaacs π-special characters 538
  45. § 41 Three arithmetical applications 548
  46. § 42 Small kernels and faithful irreducible characters 554
  47. § 43 TI-sets 567
  48. § 44 Involutions 573
  49. § 45 Groups whose Sylow-2-subgroups are generalized quaternion groups 580
  50. § 46 Perfect Frobenius complements 597
  51. Books 606
  52. Bibliography 607
  53. Index of names 614
  54. Index of subjects 617
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