Modern Algebra

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mi06bian
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245242
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Modern Algebra
Updated:
2013-12-11 21:04:00
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algebra
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Midterm terms
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  1. commutative
    • when the order does not matter
    • always get the same result no matter which order we do the binary operation
  2. group
    • a set G with a binary operation *, (G, *) such that:
    • 1. the operation * is closed and well-defined on G
    • 2. The operation * is associative on G
    • 3. There is an element e in G such that g*e = g = e*g for all elements g in G.  e is called the identity. G is non-empty.
    • 4. for each element g in G there is an element h in G such that g*h = e = h*g.  the inverse of g, g-1.
  3. Identity
    there exists an element e in G such that g * e = g = g * e for all g in G.
  4. inverse
    • g-1
    • g * h = e = h * g where h is the inverse of g.
  5. Theorem
    The integers with addition, (Z, +), is a group
  6. Theorem
    The nonzero real numbers with multiplication (R{0}, .) is a group.
  7. Theorem
    • the transformations of an equilateral triangle in the plane with composition is a group.
    • (D3,) the symmetries of the equilateral triangle
  8. Theorem 3.6 - unique identity
    • Let G be a group.
    • There is a unique identity element in G.
    • There exists only one element in G, e, such that g * e = g = g * e for all g in G.
  9. Cancellation Law
    • Let G be a group and let a, x, y exist in G.
    • Then a * x = a * y iff x = y.
  10. Corolary 3.9 - unique inverse
    • Let G be a group
    • Then each element g in G has a unique inverse in G.
    • There is only one element h such that g * h = e = h * g.
  11. Theorem 3.10 - identity is commutative
    • Let G be a group with elements g and h.
    • If g * h = e, then h * g = e.
  12. Theorem 3.11 - h and g are inverses
    • Let G be a group and g be in G.
    • Then (g-1)-1 = g.
  13. Theorem 3.13 - cyclic group of order n
    • Fore every natural number n, the set Cn with n-cyclic addition, (Cn, n) is a group.
    • Called the cyclic group of order n.
  14. Symmetry
    a transformation that takes the regular polygon to itself as a rigid object.
  15. Theorem 3.19 - symmetries of the square
    The symmetries of the square in the plane with composition form a group.
  16. dihedral group
    The symmetries of a regular n-gon form a group, denoted Dn
  17. congruent modulo
    • a = b mod n
    • two integers iff their difference is divisible by n
    • a and b are congruent modulo n if there exists an integer k st a = b+kn or a-b = kn
  18. modular addition
    • let Zn = {[a]n| a Z, [a]n = [b]n iff a = b mod n}
    • on Z is defined by [a]n [b]n = [a+b]n
  19. subgroup
    • a subgroup of a group (G, *) is a non-empty subset, H, of G along with the restricted binary operation such that (H, *|H) is a group
    • to check that a subset is a subgroup, must check all conditions of the group
  20. Theorem 3.23 - identity in subgroup
    • Let G be a group with identity element e.
    • Then for every subgroup H of G, e is in H.
  21. Theorem 3.24 - identity is a subgroup of G
    • Let G be a group with identity element e.
    • Then {e} is a subgroup of G.
  22. Theorem 3.25 - group is subgroup of itself
    • Let G be a group.
    • Then G is a subgroup of G.
  23. non-trivial
    • {e} and G are trivial
    • if H is a subgroup of G and not either of these it is non-trivial.
  24. g4
    • g*g*g*g
    • repeated operations of the binary operation to one element g in G.
    • g0=e
    • g1=g
  25. <g>
    • subset of elements of G formed by repeated operations using only g and g-1
    • <g> = {g+-1*g+-1*g+-1...}
  26. Theorem 3.27 - subgroup of G generated by g
    • Let G be a group and g be an element of G.
    • Then <g> is a subgroup of G.
  27. Theorem 3.29
    • Let G be a group and S be a subset of G.
    • Then <S> is a subgroup of G.
  28. Cyclic
    • a group G is called cyclic if there is an element g in G st <g> = G.
    • A group is cyclic if it is generated by one element
  29. Theorem 3.31 - integers under +
    The integers under addition is a cyclic group
  30. Theorem 3.33 - cyclic has cyclic subgroups
    Any subgroup of a cyclic group is cyclic
  31. Theorem 3.34 - Dn not cyclic
    • The groups Dn for n>2 are not cyclic.
    • Need flips and rotations.
  32. Finite
    a group G is finite if the underlying set is finite
  33. Infinite
    a group G is infinite if the underlying set is infinite
  34. Finitely Generated
    G = <S> for some finite subset S
  35. Theorem 3.36 - finite group is finitely generated
    Every finite group G is finitely generated.
  36. order
    • the number of elements in G, |G|.
    • the order of an element g, o(g) is the order of the subgroup that it generates.
    • o(g) = |<g>|
  37. Abelian
    • a group (G, *) is abelian iff for every pair of elements g, h in G, g*h=h*g.
    • iff its binary operation is commutative
  38. center
    • the center of a group is the collection of elements in g that commute with all elements of G
    • Z(G) = {g G | g*h = h*g V h G}
  39. Theorem 3.45
    • Let G be a group
    • Then Z(G) is a subgroup of G.
  40. Left coset of H by g
    • Let H be a subgroup of group G and g be and element of G.
    • The left coset of H by g is the set of all elements of the form gh for all h that exist in H.
    • written as gH = {gh | h  H}
  41. Lemma 3.48
    Let H be a subgroup of G and let g and g' be elements of G.  Then the cosets gH and g'H are either identical or disjoint.
  42. Lagrange's Theorem 3.49
    Let G be a finite group with subgroup H.  Then |H| divides |G|.
  43. Scholium 3.50
    Let G be a finite group with a subgroup H. Then the number of left cosets of H is equal to the number of right cosets of H.
  44. Index
    • Let H be a subgroup of a group G.  Then the index of H in G is the number of distinct left or right cosets of H. 
    • Written as [G : H]
  45. Scholium 3.51
    Let G be a finite group with a subgroup H.  Then [G : H] = |G|/|H|.
  46. Corollary 3.52
    Let G be a finite group with an element g.  Then o(g) divides |G|.
  47. Corollary 3.53
    If p is a prime and G is a group with |G| = p, then G has no non-trivial subgroups.
  48. Cartesian Product
    If A and B are sets, then we define A x B = {(a,b)| a  A and b  B}, the set of ordered pairs of elements from A and B.
  49. Theorem 3.54
    Direct Product
    • Let (G, *G) and (H, *H) be groups and define *: (G x H) x (G x H)  G x H by (g1, h1) * (g2, h2) = (g1 *G g2, h1 *H h2). 
    • Then (G x H, *) is a group, called the (direct) product of G and H.
  50. Theorem 3.56
    • Let G and H be groups.
    • Then G x H is abelian iff both G and H are abelian.
  51. Theorem 3.57
    Let G1 be a subgroup of a group G and H1 be a subgroup of a group H. Then G1 x H1 is a subgroup of G x H.
  52. Two-line notation
  53. Cycle Notation
    (123)
  54. Theorem 3.66
    Symmetric group
    • Let X be a set, let Sym(X) be the set of bijections from X to X, and let  represent composition.  Then (Sym(X), ) is a group.
    • This is the symmetric group on X.
  55. Homomorphism
    • Let (G, *G) and (H, *H) be groups and let  : G  H be a function on their underlying sets. 
    •  is called a homomorphism of the groups if for every pair of elements g1, g2  G,  (g1 *G g2) = (g1) *H  (g2).
  56. inclusion map
    • iAB : A  B is defined as follows:
    • for each element a  A, iAB(a) = a.
  57. Image and Preimage
    • Let A and B be sets and f : A  B be a function.  For any subsets S  A and T  B we define
    • 1. Imf(S) = {b  B | there exists an a  S st f(a) = b}.  We call this the image of S under f.
    • 2.Preimf(T) = {a  A | f(a)  T}.  We call this the preimage of T under f.
  58. kernel
    Let  : G  H be a homomorphism from a group G to a group H. Then the set {g  G |(g) = eH} is called the Kernel and is denoted Ker().
  59. Monomorphism
    an injective homomorphism
  60. isomorphism
    a bijective homomorphism
  61. isomorphic
    A group G is isomorphic to a group H if there exists an isomorphism,  : G  H.

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