Modern Algebra

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 Author: mi06bian ID: 245242 Filename: Modern Algebra Updated: 2013-12-11 21:04:00 Tags: algebra Folders: Description: Midterm terms Show Answers:

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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
• 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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