Pure Mathematics E2: Quotient Groups&Conjugacy

71 flashcards · Mathematics · by lakesare

1. What's the name of this operation?

   Set composition. That is, if X and Y are subsets of G then X · Y is the subset of G obtained by composing each element of X with each element of Y, in that order.

2. [Set composition] Set composition VS Cartesian product?

   Set composition Set composition X · Y is the set of elements xy.Cartesian product Cartesian product A × B is the set of all ordered pairs (a, b).

3. [Set composition] What's the result of this set composition?

4. [Set composition] We denote set composition by · and +, however what other notation could be used?

   We could use +9, ×9, ◦. The composite of subsets {1, 4, 7} + {2, 5, 8} in the group Z9.

5. [Set composition] There are 3 operations here, what's their result? el ◦ el (e.g. xy)el ◦ {} (e.g. xN) {} ◦ {} (e.g. xN · yN)

   el ◦ el = el (e.g. xy)el ◦ {} = {} (e.g. xN){} ◦ {} = {} (e.g. xN · yN) Set is infectious!

6. [Quotient groups] How do you say G/N?

   G mod N, or G over N.

7. [Normal subgroups] What would be a good name for the normal subgroup?

   Normal subgroup is "set-wise commutative".

8. [Quotient groups] How do you denote a quotient group of G by N.

   G/N.

9. [Quotient groups] Purely visually - what's the order of the quotient group?

   That's partially why the quotient group is denoted in this way!

10. Give 2 names of the G/N group.

    Quotient group Factor group

11. [Quotient groups] Why do we usually use the letters G and N when denoting the quotient group G/N?

    G - group N - normal group

12. [Cosets] Name 2 ways in which the coset { a, c } of N can be written.

    aN, cN Or Na, Nc if the subgroup is normal.

13. [Quotient groups] How can you compare forming quotient groups to division?

14. [Quotient groups] Visually imagine the Cayley table of a quotient group.

    Cayley table for the cosets of the normal subgroup {0, 3, 6} in the additive group Z9. In Cayley tables like these, it is important to denote each coset in a consistent way throughout the table.

15. [Quotient groups and normal subgroups] Normal groups VS quotient groups? N - normal subgroup; G/N - quotient group.

    normal subgroups - set-wise commutative subgroups quotient groups - { coset of N, coset of N, coset of N }

16. What kind of groups do we denote by N VS G/N?

    N - normal subgroups G/N - quotient groups

17. [Normal subgroups] Imagine this theorem visually.

18. For what groups does this theorem hold true?

    For normal subgroups.

19. [Blocking for the quotient groups] How to make the quotient group G/N apparent in a group table?

    Arrange the elements in the headings so that elements in the same coset of N are together. Blocking in the group table of Z9

20. [Blocking for the quotient groups] How are these elements colored? Blocking in the group table of Z9

    Each coset has a particular color, e.g. { 0, 3, 6 }, { 1, 4, 7 }, { 2, 5, 8 }.

21. [Blocking for the quotient groups] What does each color represent? Blocking in the group table of Z9

    Each color represents a coset.

22. [Quotient groups] What does this Cayley table of the quotient group tell us? The quotient group in question is {0, 3, 6}/Z9. Cayley table for the cosets of the normal subgroup {0, 3, 6} in the additive group Z9. In Cayley tables like these, it is important to denote each coset in a consistent way throughout the table.

    That cosets of the NORMAL subgroup form a group!

23. [Blocking for the quotient groups] Why: we can block { e, b }/S(□), but we can NOT block { e, r }/S(□).

    Because { e, b } is a normal subgroup, and { e, r } is NOT a normal subgroup. Cayley table for the cosets of the normal subgroup {0, 3, 6} in the additive group Z9.In Cayley tables like these, it is important to denote each coset in a consistent way throughout the table.

24. [Quotient groups] In G/N, what's: the identity el the inverse els?

    the identity el - N the inverse els - inverse of xN is x–1N.

25. [Quotient groups of (Z, +)] Why are all subgroups of (Z, +) normal?

    Because (Z, +) is abelian.

26. [Quotient groups] When does the G/some_subgroup quotient group exist?

    When "some subgroup" is a normal subgroup in G.

27. [Quotient groups] Visually imagine the elements of the normal subgroup.

    This is a N (normal subgroup) and G/N (quotient group) of the (Z9, +) group.

28. [Quotient groups] Visually imagine the elements of the quotient group.

    This is a N (normal subgroup) and G/N (quotient group) of the (Z9, +) group.

29. [Quotient groups of (Z, +)] What kind of group is denoted by 5Z?

30. [Quotient groups of (Z, +)] Visually imagine elements of the quotient group Z/5Z.

    The elements of Z/5Z are the cosets of 5Z in Z.We repeatedly choose an element x of Z not yet assigned to a coset and find the coset containing x, until all the elements of Z have been assigned to cosets. These five cosets between them contain every element of Z, so there are no more cosets. Hence Z/5Z has order 5.

31. [Quotient groups of (Z, +)] Looking at the group table of Z/5Z - what group is Z/5Z isomorphic to?

    (Z/5Z, +) ≅ (Z5, +)

32. [Quotient groups of (Z, +)] Why does "the quotient group R/Z" certainly exist?

    Because R is abelian, so Z is a normal subgroup.

33. [Quotient groups of (Z, +)] What exactly are the R and Z groups in "R/Z" notation?

    (R, +)/(Z, +)

34. [Quotient groups of (Z, +)] What are the elements of R/Z?

    Cosets of Z in R.

35. [Quotient groups of (Z, +)] Visually imagine quotient groups of R/Z on a line.

36. [Simple groups] Which groups are called simple?

    Group G of order >= 2 with NO normal subgroups (other than { e } and G).

37. [Simple groups] What is a classification theorem about? Much of the work was overseen by Daniel Gorenstein (1923–1982) who said in 1979 that ‘it makes more sense to view this classification as an entire field of mathematics rather than as an attempt to prove a single theorem’.

    If group is simple → it falls in one (or more) of these patterns. So apart from the 26 sporadic groups, all finite simple groups fit into patterns. The classification theorem is an enormously powerful result: many problems in group theory can be reduced to problems about simple groups or groups closely related to simple groups, allowing knowledge about simple groups to be used to solve them. By 2004 almost none of the major problems in finite group theory that were unsolved before 1980 remained…

38. [Conjugates] Visually imagine the conjugate of x.

39. [Conjugates] Name 2 ways to find the conjugate of x. E.g., find the conjugate of (1 4 3)(2 6) by (1 3 2 4 5).

    calculate - g ◦ x ◦ g–1 renaming method

40. [Conjugates] Why can we always say that x and y are "conjugates of each other"?

    y = g ◦ x ◦ g–1 y = (g–1) ◦ x ◦ (g–1 )–1

41. [Conjugates] How can we find *many* g-s? y is a conjugate of x

    Simply change the starting numbers.

42. [Conjugates] What conjugates property is visually obvious for symmetric groups, but works for all groups?

    Conjugates have the same order in any group. y = g ◦ x ◦ g–1x and y have the same order. Conjugate permutations have the same cycle structure, and therefore have the same order. Interestingly, all conjugates have the same order!

43. [Conjugates] What does this tell us?

    It is apparent that the pattern in the exercise will continue for all positive integers n. In other words, this lemma holds.

44. [Equivalence relations] Visually imagine 7 equivalence classes.

    Graph of 7 equivalence classes.

45. [Conjugates] Visually imagine the conjugacy class of a in G.

    It's H-shaped. So the conjugacy class of a in S(□) is { a, c }.

46. [Conjugates] If a and b are conjugates, what does it say about a and b cycle structures and a and b orders?

    a and b are conjugates ↔ a and b have the same cycle structure (only in Sn!!!!) → a and b have the same order (in all groups!).

47. [Normal subgroups] Name 2 equivalent definitions of the normal subgroup. Not characterisations (there are many of those), but standard definitions.

    set-wise commutative left cosets = right cosets Surprisingly, these are equivalent!

48. [Normal subgroups] What does it mean to check a subgroup normality?

    It means checking whether the subgroup is normal within the context of this group.

49. [Terminology] What's a characterisation in mathematics?

    A rational number, generally defined as a ratio of two integers, can be characterized as a number with finite or repeating decimal expansion.

50. [Normal subgroups] Name 2 characterisations of normal subgroups.

    set-wise commutative ghg–1 still belongs to H More useful for groups of ∞ and large orders. To use it to prove that a subgroup H of a group G is normal in G, we take a general element g ∈ G and a general element h ∈ H, and show that the conjugate ghg−1 belongs to H.

51. [Normal subgroups] Visually imagine the ghg–1 characterisation of a normal subgroup.

52. [Conjugates] Visually imagine why does the renaming method work. We claim that y is equal to g ◦ x ◦ g−1.

    Take a particular case (g ◦ x ◦ g–1)(3).

53. [Equivalence relations] Name 2 examples of equivalence classes.

    isomorphism equivalence classes conjugacy equivalence classes This is an equivalence relation whose equivalence classes are called conjugacy classes.congruence equivalence classes Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by an, is the set {... , a − 2n, a − n, a, a + n, a + 2n, ...}. This set, consisting of all the integers congruent to a modulo n, is called the congruence class, residue class, or simply re…

54. [Conjugates] Lagrange theorem VS conjugacy classes fact?

    Lagrange's theorem - order of subgroup | order of G conjugacy classes fact - n of els in each conjugacy class | order of G

55. [Conjugates] If H is a subgroup, what do we call g ◦ H ◦ g–1?

    The conjugate subgroup of H in G. Conjugate subgroup of H is H renamed in any way.

56. [Conjugates] State this theorem in your own words.

    Renamed subgroup is still a subgroup.

57. [Conjugates] What's a straightforward proof that subgroup H and conjugate subgroup of H are isomorphic?

58. [Normal subgroups] How do conjugacy classes relate to normal subgroups? So the conjugacy class of a in S(□) is { a, c }.

    Subgroup is normal ⇔ it's a union of conjugacy classes. A normal subgroup H of a group G: each conjugacy class of G lies either entirely inside or entirely outside H.

59. [Normal subgroups] Visually imagine normal subgroup of G with conjugacy classes of G.

    A normal subgroup H of a group G: each conjugacy class of G lies either entirely inside or entirely outside H. A subgroup H of a group G that is not normal: some conjugacy classes of G lie partly inside and partly outside H

60. [Normal subgroups] What's an interesting way to prove all 4 properties are equivalent? Although Property A was used in the original definition of a normal subgroup, any of the other three conditions could have been used in its place. We can use any of the four conditions when we wish to prove that a subgroup is normal, or show that it is not normal.

    Rather than prove the three theorems individually, we will prove that if H is a subgroup of a group G, then the following five implications hold.

61. [Conjugates] What does conjugacy mean in the context of: symmetric groups symmetry groups? For some groups it is possible to say what conjugacy 'means' in the particular context of that group.

    symmetric groups - conjugate ≡ same cycle structure symmetry groups - conjugate ≡ same geometric type a and c are of the same geometric type ≡a and c are conjugate in this S(F).The effect of the conjugate symmetry r ◦ a ◦ r−1 on the square.Conjugating a by r.

62. [Conjugates] Name 2 ways to find the conjugate r ◦ a ◦ r−1 in a symmetry group.

    apply r ◦ a ◦ r−1 to the figure apply r to the action of a (and apply it to the figure) Thus the conjugate symmetry r ◦ a ◦ r−1 is the symmetry obtained by "applying r to the action of a", as illustrated.

63. [Conjugates] What does it mean to say that a and c are of the same geometric type? a and c are of the same geometric type.

    a and c are of the same geometric type ≡ a and c are conjugate in this S(F).

64. [Conjugates] How do we know a and c symmetries are conjugate? S(□)

    By way of contrast - there is no symmetry of the square that transforms a diagram illustrating r into one illustrating s, so r and s are not conjugate in S(□).

65. [Fixed point set] What's a fixed point set?

    All points that are fixed (== not moved) by the symmetry.

66. [Fixed point set] What does this denote?

    { All points that f fixes }. A "fixed point set" of a symmetry f.

67. [Fixed point set] What's Fix(a)? S(□)

    For example, the fixed point set of a non-trivial rotational symmetry of a plane figure is the centre of rotation, if this lies in the figure, and is the empty set otherwise.

68. [Fixed point set] What's Fix(s)? S(□)

    The fixed point set of a reflectional symmetry of a plane figure is the set of all points of the figure that lie on the axis of reflection.

69. [Conjugacy classes] How would you go about finding conjugacy classes of: symmetric group Sn VS symmetry group S(F)?

    symmetric group Sn - simply partition the group by cycle structure symmetry group S(F) partition the group by cycle structure apply tricks

70. Visually imagine partitioning the group: via { cosets } via { conjugacy classes }

    via { cosets } - partitioned rectangle via { conjugacy classes } - letter H

71. [Normal subgroups] Explain these 2 strange words: kernormal conjnormal

    kernormal - normal subgroup ≡ Ker φ! conjnormal - normal subgroup ≡ union of conjugacy classes! A normal subgroup H of a group G: each conjugacy class of G lies either entirely inside or entirely outside H.