Pure Mathematics B2: Subgroups & Isomorphisms

94 flashcards · Mathematics · by lakesare

1. When is (H, ⚬) a subgroup of group (G, ⚬)?

   same operator ⚬ H ⊆ G - H is a subset of G (H, ⚬) is a group

2. Give the example of the subgroup of (S(△), ⚬). S(△) - set of all symmetries of the triangle. The group table (S(△), ⚬)

   (S+(△), ⚬) = ({ e, a, b}, ⚬)

3. Every group G has 2 subgroups: ______________ ______________

   trivial subgroup - only the identity element ({ e }, ⚬) itself - the entire group (G, ⚬)

4. What's the trivial subgroup of (Z, +)?

   ({0}, +)

5. Proper subgroup VS proper subset?

   proper subgroup - S ⊂ G proper subset - S ⊂ G

6. What stays the same in a group and subgroup?

   identity element e inverse elements h-1 for each h

7. Name 3 steps to check whether S is a subgroup of G.

   S is a subset of G Operation ◦ is the same Subgroup properties hold

8. Name 3 subgroup properties. 3 steps to check whether S is a subgroup of G:- S is a subset of G- Operation ◦ is the same- Subgroup properties hold

   Closure (usual closure) Identity (∈ subgroup) Inverses (∈ subgroup)

9. What does 3ℤ denote?

   The set of integer multiples of 3.

10. What do we mean when we say group G?

    Group (G, ◦), with ◦ clear from the context.

11. Say symmetry group of F in a notation.

    (S(F), ◦)

12. [Subgroups of the symmetry group] Name 3 ways to find subgroups of symmetry group of F. 3 ways to find subgroups of (S(F), ◦).

    S+(F) - all direct symmetries Modify F to restrict its symmetry Fix vertex/edge of the figure. Symmetries of the tetrahedron that fix the vertex at location 4.

13. [Subgroups of the symmetry group] Write down a subgroup of S(□) by listing the symmetries of the figure F'. Let F' be the modified square.

    Subgroup of S(□) is { e, b, r, t }.

14. [Subgroups of the symmetry group] What's important to remember when you are finding the subgroup of S(tet) consisting of the symmetries that fix the edge joining the vertices at locations 1 and 2.

    This subgroup consists of all symmetries: that either fix the vertices at location 1 and 2 or interchange them!

15. [Powers of a group element] What do we mean by powers of a group element x ∈ (G, ◦)?

    x ◦ x ◦ x... (what happens on repeated composition)

16. [Powers of a group element] What's the x0? x ∈ (G, ◦)

    x0 = e.

17. [Powers of a group element] What's the x3? x ∈ (G, ◦)

    x3 = x ◦ x ◦ x.

18. [Powers of a group element] What's the x–3? x ∈ (G, ◦)

    x–3 = x–1 ◦ x–1 ◦ x–1.

19. [Powers of a group element] Find c0.

    x0 = e.

20. [Powers of a group element] Find a2.

    a2 = b.

21. Name 2 notations for group operations.

    Multiplicative notation - x ◦ x ◦ x = x3 for (G, ◦) Additive notation - x + x + x = 3x for (G, +)

22. When do we use Multiplicative notation VS Additive notation?

    Multiplicative notation - for theorems and proofs in group theory (and for groups in which the operator is some kind of multiplication). Theorems, proofs and general discussions about group theory are normally expressed in multiplicative notation, both in this module and in mathematics in general. If you want to apply them to additive groups, then you have to translate them into additive notation.We always use multiplicative notation for groups in which the binary operation is some kind of mult…

23. Give an example for why we introduce additive notation.

    It would be confusing to denote x + x + x by x3 in a group (R, +).

24. What's order of the group VS order of the group element?

    order of the group - size of the set G. |G| order of the group element - how many times we need to compose x with itself so that it returns to e. xn = e

25. [Order of the group element] When does the element x have finite VS infinite order?

    xn = e x has finite order - n exists. x has infinite order - such n doesn't exist.

26. [Order of the group element] What's the order of a?

    Order of a is 4.

27. [Order of the group element] What's the order of 2?

    Order of 2 is infinite.

28. [Order of the group element] Find the order of the element a of S(△).

    Order of a is 3.

29. [Order of the group element] What can you say about the order of: x ∈ finite group x ∈ ∞ group?

    x ∈ finite group - always has finite order. x ∈ ∞ group - can have either finite or ∞ order.

30. [Order of the group element] What order does the identity element have?

    Identity element has the order 1. e1 = e, so the smallest positive integer n such that en = e is 1.

31. [Order of the group element] What order does the self-inverse element have?

    Self-inverse element has the order 2. If x = x−1 then (by composing each side by x) we have x2 = e. This tells us that, provided x1 = e, the smallest positive integer n such that xn = e is 2.

32. [Order of the group element] Visualize how a and a–1 always have the same order.

    a ○ a ○ a ○ a = e c ○ c ○ c ○ c = e

33. [Order of the group element] What pattern do we have when the order of x is finite VS order of x is ∞?

    Here is a pattern where a has finite order 4: Here is a pattern where 2 has infinite order:

34. [The cycle of powers of x] What does the cycle of powers of x ∈ (G, ◦) look like visually?

    The effect of repeatedly composing a group element x of order n with itself.

35. [The cycle of powers of x] Where do x and x-1 appear visually in the cycle of powers of x?

    To the left and to the right of e.

36. [Finding the orders of each group element] What does this image mean?

    the order of e is 1 the order of self-inverse elements is 2 the order of x and x–1 have the same

37. [Finding the orders of each group element] How can you use the cycle of powers of x to find the order of each group element?

    You can use the cycle of x to find the powers of any group element that appears in the cycle. Moving round the cycle of powers of a in S(□) in the reverse direction. Moving round the cycle of powers of a in S(□) two places at a time.

38. How is the "the subset〈x〉 of the group G" notation read?

    The subset generated by x of the group G.

39. [Subset of the group generated by x] What's meant by "subset of G generated by x" 〈x〉? x is an element of the group (G, ◦).

    { all powers of x }.

40. [Subset of the group generated by x] Find the subset〈a〉of the group S(△).

    〈a〉= { e, a, b }

41. [Subset of the group generated by x] Find the subset〈2〉of the group (R*, ×).

    〈2〉= { ... , 2-1, 20, 21, ... } Don't forget the negative powers!

42. How can you imagine <x> visually?

43. What's a cyclic subgroup?

    Every <x>.

44. The order of x VS the order of <x>?

    order of x - power of x (xn = e). The smallest positive power of x that equals the identity.order of <x> - size of the set. The number of elements in the set. The order of x = the order of <x>! This theorem connects the two uses of the word "order".

45. Give the example of a finite group to show that order of x = order of <x>.

46. Give the example of an infinite group to show that order of x = order of <x>.

47. What's the identity element of S(◯)?

    r0

48. What's a subgroup generated by qθ?

    <qθ> = { r0, qθ }

49. In the context of disk symmetries - what does r1 mean?

    The rotation through 1 radian.

50. What would it mean for (R*, ×) to be a cyclic group?

    There would need to exist a real number that generates all other real numbers on self-multiplication. such x ∈ G that <x> = R*.

51. How does knowing that "all subgroups of a cyclic group are cyclic" make finding all subgroups of a group easier?

    It means that we simply need to find all <el>, and that will be ALL subgroups!

52. What's a cyclic group G?

    A group generated by a single element: <x> = G. If there is such x ∈ G that <x> = G.

53. Why is (Z6, +6) a cyclic group?

    Because <1> = Z6. And because <5> = Z6.

54. Give an example of an ∞-order group that's a cyclic group?

    (Z, +) because <1> = Z.

55. Why every cyclic group is abelian?

    Because if a group is cyclic, then we can express both elements as aj and ak.

56. How should you imagine visually cyclic groups?

    The effect of repeatedly composing a group element x of order n with itself.

57. How are cyclic groups and cyclic subgroups connected?

58. How do we know (Zn, +n) is a cyclic group?

    Because <1> = Zn. 1 generates the entire group. (Z6, +6)

59. How would you approach finding the order of an element in (Zn, +n)?

    order = n / hcf(n, el)

60. Why do the orders of each element in (Z5, +5) look like this?

    Because hcf(prime, el) is always 1! order = n / hcf(n, el)

61. Which elements of (Zn, +n) are generators of (Zn, +n)?

    All elements of Un! We know order(el) = n/hcf(el, n). We know order(el) = n iff the el is a generator. Therefore, el is generator iff hcf(el, n) = 1.

62. How would you find all the generators of (Z10, +10)?

    Simply list all Un!

63. Imagine visually, for (Zn, +n): subgroup generated by the coprime number, VS subgroup generated by the number that divides n?

64. Visually imagine why every element in (Zp, +p) has order p.

65. What do <x> VS [[x]] denote?

    <x> - set of all powers of x. [[x]] - the equivalence class of x. This idea is associated only with equivalence relations, not with relations in general.

66. How do we create this colored pattern from the Cayley table? The pattern of the Cayley tables Cayley tables

    Just color each element a fixed color. We don't care about the color, we care about whether we have stripes.

67. What would you need to do to turn Figure 32 into Figure 31?

    Swap the rows 3 and 4 AND swap the columns 3 and 4.

68. Why do we have to rearrange both 3-4 columns and 3-4 rows in the same way?

    Because intermediate tables lose its properties, e.g. the fact that identity elements are symmetric across the diagonal.

69. How do we know one pattern be rearranged into the other pattern?

    No matter how we rearrange the borders of the Cayley tables, the 4 cells on the main diagonal will contain 4 occurrences of the identity element.

70. What's an interesting fact about these Cayley table patterns?

    All groups of order 4 can be rearranged into these 2 patters!

71. What type of object is an isomorphism?

    Isomorphism is a function.

72. How does this read?

    (G, ◦) is isomorphic to (H, *).

73. What do we mean when we write this? (This is not the function notation!)

    We mean φ: G → H, but we indicate binary operations too.

74. Informally - when the 2 groups are isomorphic to each other?

    When 2 groups have exactly the same structure (even when their elements and binary operations are different).

75. What's the name of this function?

    An isomorphism.

76. Explain the φ(x ◦ y) = φ(x) ∗ φ(y) requirement visually.

    * ·· °* · .*· •.•☆ · · ✯ *° ✷ * *. * ✸ • * °☆✵* • • °*. ✵. ··✧ · · • ✯ * °. . . ° ° • • *° • . • .* ✶ ·* **• ° • ✸ * *✧ °✵ · * ★✯. •* ✧ ✯°*·• · • . ° . •° ·• ☆ * **

77. How do we know that relation "is isomorphic to" is an equivalence relation?

    Because "is isomorphic to" relation satisfies: reflexivity symmetry and transitivity properties.

78. What exactly do we mean by the isomorphism class?

    Isomorphism class = equivalence class where all groups are isomorphic to each other.

79. What are the names for these isomorphism classes: 2 self-inverse els - ______________ 4 self-inverse els - ______________ There are exactly 2 isomorphism classes of groups of order 4.

    2 self-inverse els - C4, the cyclic group of order 4 4 self-inverse els - V, Klein four-group Felix Klein

80. What goes as a proof that φ is onto?

81. Name 2 ways to show that two groups are isomorphic to each other.

    Find an actual isomorphism φ from one group to another. Rely on known group properties.

82. Why is the ∗ operation called "unusual" here?

    Because (R, +) is more traditional, and it's easier to understand why (R, +) is a group.

83. Name 2 examples of properties of isomorphisms that arise simply because two groups have the same structure.

    φ matches identities φ matches inverses φ matches powers φ matches the order of the element φ matches subgroups φ matches whether the group is abelian φ matches whether the group is cyclic

84. What does "applying the isomorphism" mean?

    Taking φ of both sides.

85. What does φ(g3) = 3φ(g) even refer to?

    It's the property that isomorphism matches powers, but in additive notation.

86. What's the approximate strategy: to show that 2 groups are isomorphic VS to show that 2 groups are NOT isomorphic?

    isomorphic - find φ, or use facts about isomorphism classes. NOT isomorphic - rely on the properties of isomorphisms!

87. Imagine visually: all cyclic groups of the same order are isomorphic to each other.

    The cycle of powers of a in (S+(□), ◦) and the cycle of powers of 1 in (Z4 , +4).

88. What's the isomorphism for all cyclic groups of order 3? All cyclic groups of the same order are isomorphic to each other.

    Each power of the generator a of the 1st cyclic group matches up with the corresponding power of the generator b of the 2nd group. Example:

89. What does the notation Cn signify?

    Cyclic group of order n (they all have the same structure!).

90. Give an example of actual cyclic groups of order 4 being isomorphic to each other.

    Direct symmetries of the square (S+(□), ◦) Addition modulo 4 (Z4 , +4) The cycle of powers of a in (S+(□), ◦) and the cycle of powers of 1 in (Z4 , +4).

91. What would you have to do to find different isomorphisms of two cyclic groups?

    Use different generators!

92. How many isomorphisms can there be between a cyclic group with 2 generators, and a cyclic group with 2 generators?

    At most 4 isomorphisms, but can be fewer than 4!

93. How is this theorem nice and significant to us?

    It tells us all cyclic groups are similar to (Zn, +n)! Finally, you met the powerful concept of isomorphism, which links groups that have the same structure and which can therefore be considered in a sense to be 'the same group'.The idea of isomorphism is extremely important in group theory. As you saw above, it allows us to prove results about the structure of one group, or one family of groups, and know that these results also apply to many other groups – namely, to all groups isomorphic to t…

94. Imagine visually - how do we approach checking whether some subset is a subgroup?

    We can obtain a Cayley table for this subset by deleting all other rows and columns. And then check the 3 SG (subgroup) properties.