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Fichas

Group Theory

Sophia Martin Inglés 66 fichas

Group Theory

Fichas del conjunto (66)

  1. 1Lagrange's TheoremLagrange's Theorem
  2. 2SubgroupSubgroup
  3. 3CosetCoset
  4. 4ConjugateConjugate
  5. 5Normal subgroupNormal subgroup
  6. 6Quotient groupQuotient group
  7. 7HomomorphismHomomorphism
  8. 8Kernel, imageKernel, image
  9. 9Injective, surjective iff Isomorphism definitionInjective, surjective iff Isomorphism definition
  10. 10Natural quotient homomorphismNatural quotient homomorphism
  11. 11First isomorphism theoremFirst isomorphism theorem
  12. 12Non-trivial group G is simple ifNon-trivial group G is simple if
  13. 13Subgroup of G generated by set SSubgroup of G generated by set S
  14. 14Finitely generated groupFinitely generated group
  15. 15- Word - Elementary contraction - Reduced word- Word - Elementary contraction - Reduced word
  16. 162 propositions about reduced words2 propositions about reduced words
  17. 17down diagonal arrowsdown diagonal arrows
  18. 18If a, b, y are words st a/b\y thenIf a, b, y are words st a/b\y then
  19. 19[w][w]
  20. 20Free group on set SFree group on set S
  21. 21Every non-trivial element of a free group...Every non-trivial element of a free group...
  22. 22Any group is isomorphic to...A quotient of a free group
  23. 23Universal mapping propertyUniversal mapping property
  24. 24Normal closureNormal closure
  25. 25<>&lt;&gt;
  26. 26PresentationPresentation
  27. 27Canonical presentationCanonical presentation
  28. 28Iff about homomorphismsIff about homomorphisms
  29. 291st isomorphism theorem1st isomorphism theorem
  30. 30Let H be a subgroup and N a normal subgroup. HN =Let H be a subgroup and N a normal subgroup. HN =
  31. 312nd isomorphism theorem2nd isomorphism theorem
  32. 323rd isomorphism theorem3rd isomorphism theorem
  33. 33_ G And proposition about it_ G And proposition about it
  34. 34One-to-one correspondence involving _ HOne-to-one correspondence involving _ H
  35. 35Theorem about A5Theorem about A5
  36. 36Classification of Finite Simple GroupsClassification of Finite Simple Groups
  37. 37Composition seriesComposition series
  38. 38Jordan-Holder theoremJordan-Holder theorem
  39. 39SolvableSolvable
  40. 40BurnsideBurnside
  41. 41Feit-ThompsonFeit-Thompson
  42. 423 theorems about solvability of finite G3 theorems about solvability of finite G
  43. 43Is symmetric group Sn solvable?Not for n > 4
  44. 44Derived subgroupDerived subgroup
  45. 45Facts about G'Facts about G'
  46. 46Derived seriesDerived series
  47. 47Derived lengthDerived length
  48. 48Finite derived length is equivalent toFinite derived length is equivalent to
  49. 49Internal semi-direct productInternal semi-direct product
  50. 50Conjugation restrictionφh(g) = hgh^-1
    Conjugation by h is an automorphism of G    Conjugation restriction
  51. 51Group of semi-direct product pairsGroup of semi-direct product pairs
  52. 52ExtensionExtension
  53. 53Short exact sequenceShort exact sequence
  54. 54SplitSplit
  55. 55Homomorphisms which split sequenceHomomorphisms which split sequence
  56. 56p-groupp-group
  57. 57Sylow p-subgroupSylow p-subgroup
  58. 58Sylow's first theorem(Assuming G is finite and p prime. We may then uniquely write |G| = p^a m with a > 0, m > 1 and p doesn't divide m)
    There's a sylow p subgroup of G
  59. 59Sylow's 2nd and 3rd theorem(Assuming G is finite and p prime. We may then uniquely write |G| = p^a m with a > 0, m > 1 and p doesn't divide m)Sylow's 2nd and 3rd theorem
  60. 60Group of order pqGroup of order pq
  61. 61Remark about semi-direct productRemark about semi-direct product
  62. 62Useful test for normal subgroupIff its a union of conjugacy classes
  63. 63IndexNumber of left cosets
    [G:H] = |G/H|
    H normal if index =2
  64. 64Normal subgroup if and only ifNormal subgroup if and only if
  65. 65Group of order pq cyclic ifp > q and q doesn't divide p-1
  66. 662 if conditions for group to be solvable- Solvable if finite and odd order
    - Solvable if order < 60 (as smallest solvable group is A5 which has order = 60)

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Fichas sobre Group Theory, creado por Sophia Martin el 24/04/2023.
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Sophia Martin
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