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Flashcards

Groups

Melissa Quail Part 1A Inglês 67 flashcards

Flashcards do conjunto (67)

  1. 1Definition of a groupPage 2
  2. 2Proof of other group properties from definitionPage 3
  3. 3Subgroup criterion + proofPage 5
  4. 4Subgroups of (Z, +, 0) are precisely the subsets nZPage 5
  5. 5D_2n is a group with 2n elementsPage 6
  6. 6Definition of homomorphism and proof of basic propertiesPage 8
  7. 7Definition of image and kernel of homomorphismPage 8
  8. 8Image and Kernel are subgroups in respective groupsPage 8
  9. 9Injectivity of homomorphism condition and proofPage 9
  10. 10A cyclic group is isomorphic to some Cn or (Z, +, 0)Page 9
  11. 11Mobius group definition and proof of group structurePage 11
  12. 12Every Mobius transformation is a composition of z+b, az and 1/z (in fact only two of these - prove)Page 12/ example sheet
  13. 13*Mobius transformation with >= 3 fixed points is identityPage 13
  14. 14*There is a unique mobius transformation mapping any 3 distinct complex numbers to and 3 distinct complex numbersPage 13
  15. 15Conjugacy is an equivalence relationPage 14
  16. 16*Any non-identity Mobius transformation has 1 or 2 fixed points (what are they conjugate to?)Page 14
  17. 17Definition of a 'circle' in CPage 14
  18. 18Mobius transformations send 'circles' to 'circles'Page 15
  19. 19An invertible function preserves cross ratios iff it is a Mobius transformationPage 15
  20. 20Condition for 4 points to lie on a circle in C, based on cross ratioPage 15
  21. 21Action definition (left and right)Page 16
  22. 22Action is equivalent to a homomorphismPage 16
  23. 23Cayley's theoremPage 17
  24. 24Definitions of orbit, stabiliser, kernel, transitive, faithful with regard to group actionsPage 17
  25. 25Stabiliser is a subgroup, orbits partition XPage 17
  26. 26Set of orbits notation for left and right actionsPage 18
  27. 27Regular actionPage 19
  28. 28Coset definitionPage 19
  29. 29Lagrange's TheoremPage 20
  30. 30Fermat-Euler TheoremPage 21
  31. 31*Orbit-Stabiliser TheoremPage 22
  32. 32*Cauchy's TheoremPage 23
  33. 33Conjugation action and associated terminologyPage 24
  34. 34Conjugation of a subgroup produces another subgroup (i.e. G acts on its subgroups by conjugation)Page 24
  35. 35Natural bijection between left and right cosetsExample sheet
  36. 36Quaternion matricesPage 25
  37. 37Chinese Remainder TheoremPage 25
  38. 38Direct product theoremPage 26
  39. 39*Classification of groups up to size 8 (and size 2p)Page 27 (and example sheet)
  40. 40Normal subgroup iff gH=Hg for all gPage 29
  41. 41Subgroup is normal iff it is the kernel of some homomorphismPage 29 and example sheet
  42. 42*Isomorphism theoremPage 31
  43. 43Use a tetrahedron to show that S4/K is isomorphic to S3 for some normal subgroup KPage 33
  44. 44Difference between Sn and Sym(X)Page 33
  45. 45Disjoint cycles commute and every permutation can be 'uniquely' written as a composition of disjoint cycles in which each 1 - n appears exactly oncePage 35
  46. 46Order of a permutationPage 35
  47. 47Every permutation is a product of transpositionsPage 36
  48. 48Sign of a permutation is a well-defined homomorphismPage 36
  49. 49Conjuagation and conjugacy classes in SnPage 37
  50. 50Size of stabilisers and conjugacy classes in SnPage 38
  51. 51Conjugation and conjugacy classes in AnPage 40
  52. 52A5 is simplePage 41
  53. 53Definitions of common matrix groupsPage 42
  54. 54How does GLn act on Mnxn ?Page 42
  55. 55Jordan Normal FormPage 43
  56. 56Homomorphism from SL2(C) to M is surjective with Kernel + or - IPage 44
  57. 57Projective Special Linear GroupPage 45
  58. 58*A matrix is in O(n) iff it preserves dot productPage 46
  59. 59Reflections and proof that Ra is in O(n)Page 46
  60. 60Conjugation of reflectionPage 47
  61. 61Every element of O(n) is a rotation or reflectionPage 48
  62. 62Every element of O(2) is a composition of at most 2 reflectionsPage 49
  63. 63For every A in SO(3) there is a unit vector v such that Av=vPage 49
  64. 64Every matrix in SO(3) is a rotationPage 49
  65. 65Every matrix in O(3) is a composition of at most 3 reflectionsPage 50
  66. 66Every group of order p^2 is abelianExample sheet 2
  67. 67Show that if a normal subgroup N of O(2) contains a reflection that N = O(2)Example sheet 4

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Part 1A FlashCards sobre Groups, criado por Melissa Quail em 09-01-2019.
Criado por
Melissa Quail
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