GT4. Normal Subgroups and Quotient Groups 
GT4. Normal Subgroups and Quotient Groups
by Robert Donley
Video Lecture 9 of 83
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Date Added: March 9, 2015

Lecture Description

Abstract Algebra: We define normal subgroups and show that, in this case, the space of cosets carries a group structure, the quotient group. Example include S3, the modular integers, and Q/Z.

U.Reddit course materials available at ureddit.com/class/23794/intro-to-group-theory Master list at mathdoctorbob.org/UReddit.html

Course Index

  1. Mystery Division Problem
  2. GT1. Definition of Group
  3. GT1.1. Example of Group Inverse
  4. Order 2 Elements in Finite Group
  5. Example of Group Cancellation Law
  6. GT2. Definition of Subgroup
  7. Example of Group: GL(2, R) (1 of 3)
  8. GT3. Cosets and Lagrange's Theorem
  9. GT4. Normal Subgroups and Quotient Groups
  10. GT5. Index 2 Theorem and Dihedral Groups
  11. Example of Group: GL(2,R) (2 of 3)
  12. Example of Group: GL(2, R) (3 of 3)
  13. GT6. Centralizers, Normalizers, and Direct Products
  14. GT7. The Commutator Subgroup
  15. GT8. Group Homomorphisms
  16. GT9. Group Isomorphisms
  17. Example of Group Isomorphism
  18. GT10. Examples of Non-Isomorphic Groups
  19. GT11. Group Automorphisms
  20. GT11.1. Automorphisms of A4
  21. GT12. Aut(Z/n) and Fermat's Little Theorem
  22. GT12.1. Automorphisms of Dihedral Groups
  23. GT13. Groups of Order 8
  24. GT14. Semidirect Products
  25. GT15. Group Actions
  26. GT16. Cayley's Theorem
  27. GT16.1 Examples of Cayley's Theorem
  28. GT17. Symmetric and Alternating Groups
  29. Order 12 Subgroups in S5
  30. GT17.1. Permutation Matrices
  31. |Z(G)| for |G|=pq
  32. GT18. Conjugacy and The Class Equation
  33. Class Equation for Dihedral Group D8
  34. GT18.1. Class Equation for Dihedral Groups
  35. GT18.2. A_n is Simple (n ge 5)
  36. GT19. Cauchy's Theorem
  37. GT20. Overview of Sylow Theory
  38. GT20.1. Sylow Theorems - Proofs
  39. GT20.2 Sylow Theory for Simple 60
  40. Sylow Theory for Order 12 Groups 1
  41. Sylow Theory for Order 12 Groups 2
  42. Simple Group 168 - Sylow Theory - Part 1
  43. Simple Group 168 - Sylow Theory - Part 2
  44. GT21. Internal Products
  45. GT22. The Fundamental Theorem of Finite Abelian Groups
  46. GT23. Composition and Classification
  47. RNT1.1. Definition of Ring
  48. RNT1.2. Definition of Integral Domain
  49. RNT1.2.1. Example of Integral Domain
  50. RNT1.2.2. Order of a Finite Field
  51. RNT1.3. Ring Homomorphisms
  52. RNT1.4. Ideals and Quotient Rings
  53. RNT1.4.1. Example of Quotient Ring
  54. RNT2.1. Maximal Ideals and Fields
  55. RNT2.1.1. Finite Fields of Orders 4 and 8
  56. RNT2.2. Principal Ideal Domains
  57. RNT2.3. Euclidean Domains
  58. RNT2.3.1. Euclidean Algorithm for Gaussian Integers
  59. RNT2.4. Gaussian Primes
  60. RNT2.5. Polynomial Rings over Fields
  61. RNT2.5.1. Euclidean Algorithm for Z/3[x]
  62. RNT2.6.1. Gauss' Lemma
  63. RNT2.6.2. Eisenstein's Criterion
  64. FIT1.1. Number Fields
  65. FIT1.2. Characteristic p
  66. FIT2.1. Field Extensions
  67. FIT2.2. Simple Extensions
  68. FIT2.2.1. Example: Cubic Extension
  69. FIT2.2.2. Example: Quartic Extension
  70. FIT2.3.1. Algebraic Numbers
  71. FIT2.3.2. Cardinality and Transcendentals
  72. FIT2.3.3. Algebraic Extensions
  73. FIT3.1.1. Roots of Polynomials
  74. FIT3.1.2. Roots of Real Polynomials
  75. FIT3.1.3. Example of Splitting Field
  76. FIT3.1.4. Factoring Example: Artin-Schreier Polynomials
  77. FIT3.2.1. Cyclotomic Polynomials
  78. FIT3.2.2. Mobius Inversion Formula
  79. FIT4.1. Galois Group of a Polynomial
  80. FIT4.2. Automorphisms and Degree
  81. FIT4.3. Galois Correspondence 1 - Examples
  82. FIT4.3.1. Galois Group of Order 8
  83. FIT4.3.2. Example of Galois Group over Finite Field

Course Description

Includes course on Group Theory (problems and solutions at website) and Ring Theory, and Field Theory. For Prerequisites on proofs and sets, see the Math Major Basics course.

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