MTH633 Quiz 2 Data

DATA for QUIZ#2---MTH633---MODULE # 77 to 97

1. A translation of the plane R2 in the direction of the vector (a, b) is a function f :R2→ R2 defined by f (x, y) = (x + a, y + b). 2. The composition of this translation with a translation g in the direction of (c, d) is the function f g:R2→ R2, where f g(x, y) = (x + c + a, y + d + b) This is a translation in the direction of (c + a, d + b) 3. The set of all translations in R2 forms an abelian group, under composition. 4. A translation of the plane R2 in the direction of the vector (0, 0) is an identity function 1R2:R2→ R2 defined by 1R2(x, y)=(x+0, y+0)=(x, y). 5. The inverse of the translation of the plane R2 in the direction of the vector (a, b) is an inverse function f - 1:R2→ R2defined by f -1(x, y) = (x -a, y -b) such that f f-1(x, y)=(x, y)=f-1f(x, y) 6. The inverse of the translation in the direction (a, b) is the translation in the opposite direction (−a,−b). 7. An isomorphism ϕ: G→G', if one exists, is an example of structure-relating map. 8. If(G,・) and(H,*) are two groups, the function f:G→H is called a group homomorphism if f(a・b)=f(a)*f(b) for all a,b∈G. 9. We often use the notation f : (G, ・) → (H, *) for such a homorphism. Many authors use morphism instead of homomorphism.

10. A group isomorphism is a bijective group homomorphism, If there is an isomorphism between the groups (G,・) and (H,*), we say that (G,・) and (H,*) are isomorphic and write (G,・) ≅(H,*)
11. Let ϕ: G→G' be a group homomorphism of G onto G' then if G is abelian, then G' must be abelian.
12. The function f : Z→ Zn , defined by f (x) = [x] is the group homomorphism (EXAMPLE P#14)
13. The function f : R→ R+, defined by f (x) = ex is a homomorphism, for if x, y ∈R, then f(x + y) = ex+y= exey= f (x) f (y). (EXAMPLE P#15)
14. Now f is an isomorphism, for its inverse function g:R+→R is ln x. The additive group R is isomorphic to the multiplicative group R+. The inverse function g is also an isomorphism.
15. Let Sn be the symmetric group on n letters and let ϕ:Sn→ ℤ2be defined by ϕ(ς) = 0 if ς is an even permutation, ϕ(ς)= 1 if ς is an odd permutation.
16. If ς and μ can both be written as a product of an odd number of transpositions then ςμ can be written as the product of an even number of transpositions.
17. Let ϕ:G → H be a group morphism and let eG and eH be the identities of G and H, respectively. Then (i) ϕ(eG) = eH
18. Let ϕ:G → H be a group morphism and let eG and eH be the identities of G and H, respectively. Then (ii) ϕ(a−1) = ϕ(a)−1for all a ∈G.
19. Let ϕ be a mapping of a set X into a set Y and let A⊆ X and B ⊆Y. The image ϕ[A] of A in Y under ϕ is {ϕ(a) |a ε A}. The set ϕ[X] is the range of ϕ. The inverse image ϕ-1[B] of B in X is {x ε X | ϕ(x) ε B}.
20. Let ϕ be a homomorphism of a group G into a group G'. If H is a subgroup of G then ϕ[H] is a subgroup of G'. If K' is a subgroup of G' then ϕ-1[K'] is a subgroup of G.
21. If N is a normal subgroup of a group G, the left cosets of N in G are the same as the right cosets of N in G.
22. Let h be a homomorphism from a group G into a group G’. Let K be the kernel of h. Then a K = {x in G | h(x) = h(a)} = h-1[{h(a)}] and also K a = {x in G | h(x) = h(a)} = h-1[{h(a)}]
23. When h is a homomorphism of groups then the cosets(left or right) of the kernel of h are the equivalence classes of this equivalence relation.
24. If h: X →Y is any map of sets Then h defines an equivalence relation ~h on X by: x ~hy ⇔h(x) = h(y)
25. If ϕ:G→G' is a group morphism, then kernel of ϕ is defined to be the set of elements of G that are mapped by f to the identity of G'. That is Ker f={g∈G|f(g)=e'}
26. Let ϕ:G→G' be a group morphism Then ϕ is injective if and only if Ker ϕ={e}
27. To Show ϕ:G → G' is an Isomorphism Step 1 Show ϕ is a homomorphism. Step 2 Show Ker(ϕ)= {e}. Step 3 Show ϕ maps G onto G'.
28. Let G be a group with subgroup H. The right cosets of H in G are equivalence classes under the relation a≡bmodH, defined by ab−1∈H
29. Let G be a group ,we can define the relation L on G so that aLb if and only if b−1a∈H. This relation L is an equivalence relation and the equivalence class containing a is the left coset aH={ah|h∈H}
30. The right cosets of H=A3 in S3 are H={(1),(123),(132)};H(12)={(12),(13),(23)} The left cosets of H=A3 in S3 are H={(1),(123),(132};(12)H={(12),(23),(13)}
31. The right cosets of K={(1),(12)} in S3 are K={(1),(12)};K(13)={(13),(132)};K(23)={(23),(123)} The left cosets of K={(1),(12)} in S3 are K={(1),(12)};(23)K={(23),(132)};(13)K={(13),(123)}
32. A subgroup H of a group G is called a normal subgroup of G if g−1hg ∈H for all g ∈G and h ∈H.
33. Also Hg=gH for all g∈G if and only if H is a normal sub group of G.
34. If N is a normal subgroup of (G, ·), the set of cosets G/N = {Ng|g∈G} forms a group (G/N, ·), where the operation is defined by (Ng1) · (Ng2) = N(g1· g2) then

•  This group is called the quotient group or factor group of G by N.
•  Multiplication of cosets is well defined
•  The operation is associative
•  The identity is Ne=N
•  The inverse of Ng is Ng−1
•  Ng−1·Ng=N

Hence (G/N,·) is a group.

37. The relation a ≡ b mod H is an equivalence relation on G.
38. Let (G,·) be a group with subgroup H. For a,b∈G, we say that a is congruent to b modulo H and write a≡b mod H if and only if ab−1∈H

•  If (ℤ,+) is abelian then every subgroup is normal

[a]+[b]=[a+b]

•  (ℤn,+) is a cyclic group with 1 as a generator.

40. Let K be the kernel of the group morphism f :G → H Then G/K is isomorphic to the image of f and the isomorphism ψ: G/K → I m f is defined by ψ(Kg) = f(g).

•  This result is also known as the first isomorphism theorem.
•  The function ψ is a morphism because ψ(Kg1Kg2) = ψ(Kg1)ψ(Kg2).
•  If ψ(Kg) = eH then f(g) = eH and g ∈K.

Hence the only element in the kernel of ψ is the identity coset K and ψ is injective.

41. The equivalence class containing a can be written in the form Ha = {ha|h∈H} It is called a right coset of H in G. The element a is called a representative of the coset Ha.
42. Let 𝜑 be a homomorphism function from group (G, *) to group (G‘, .) Then (Ker𝜑,*) is a normal subgroup of (G,*)
43. A subgroup H of a group is a normal subgroup if gH=Hg for ∀𝑔∈G.
44. Any subgroups of Abelian group are normal subgroups.
45. If Hg=gH then there exists h'∈H such that hg=gh' for ∀ℎ∈𝐻.
46. Hg=gH, it does not imply hg=gh.

Q47. Let H be a subgroup of a group G. When is (a H) (b H) = a b H?

•  This is true for abelian groups but not always when G is non abelian.
•  Consider S3: Let H = {ρ0, μ1}. The left cosets are {ρ0, μ1}, {ρ1, μ3}, {ρ2, μ2}.
•  If we multiply the first two together, then {ρ0, μ1}, {ρ1, μ3} = {ρ0ρ1, ρ0μ3,μ 1ρ1, μ 1μ3} = {ρ1, μ3,μ2, ρ 2}
•  This has four distinct elements, not two!

48. Let (H,*) be a normal subgroup of the group (G,*). (G/H,⨂)is called quotient group, where the operation ⨂ is defined on G/H by Hg1⨂Hg2= H(g1*g2). If G is a finite group, then G/H is also a finite group and |G/H|=|G|/|H|.
49. The product of two sets is define as follow SS’ = {xx’|x∈S and x’∈S}
50. {aH|a∈G, H is normal} is a group denote by G/H and called it factor groups of G.
51. A mapping f: G→G/H is a homomorphism and call it canonical homomorphism.
52. Consider S3: Let H = {ρ0, ρ1, ρ2}. The left cosets are {ρ0, ρ1, ρ2}, {μ1, μ2, μ3} The cosets of {ρ0, ρ1, ρ2} with this binary operation form a group isomorphic to ℤ2.
53. There is a natural map from S3 to {{ρ0, ρ1, ρ2}, {μ1, μ2, μ3}} that takes any element to the coset that contains it. This gives a homomorphism called the canonical homomorphism.
54. Let H be a subgroup of a group G Then H is normal if and only if (a H )( b H) = (a b) H, for all a, b in G
55. Lagrange’s Theorem shows that if there is a subgroup H of a finite group G, then the order of H divides the order of G.
56. The converse of Lagrange’s theorem holds for abelian groups.