Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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16 ADVANCED ABSTRACT ALGEBRA Lemma 2: Let be a homomorphism of G into , then 1. 2. (e) = , the identify element of n 3. f ( x ) = f ( x) Proof : (1) is proved above n ( ) ∀x ∈G −1 −1 bg d i bgd i (2) e = f e = f x x = f x f x − ( f ( x )) e = f ( x ) −1 1 ( ) f ( x ) f ( x ) in G 1 − −1 −1 −1 ( f ( x) ) = ef ( x ) = f ( x ), ∀x ∈ G Example 4: G = GL (2, R): group of nonsingular 2 × 2 matrices over reals and R* be the group of nonzero real number under multiplication. Then Then f: G = GL (2, R) R* defined by A (A) = det A (AB) = det (AB) = det A det B = (A) (B) ∀ e G( A f f Hence is a homomorphism ⇔ A ∈ SL( 2, R), the group of nonsingular 2×2 matrices over R, whose determinant is 1. Therefore, ker Theorem 4: (Fundamental Theorem of Group Homomorphism) Lef : G be a group homomorphism with K = ker . Then G ≅ k f ( G) G i.e. ≅ Image ( ) ( f ) ker f ( ≅ : Isomorphic, when is homomorphism, 1–1 and onto).
UNIT-I 17 Proof: Consider the diagram f G where The above diagram should be completed to G f f ( g ) = Kg GφG φ f K We shall use g f ( g) f Kg to complete the previous diagram. f kg = f g coset Kg G s ∀ ∈ Define ( ) ( ) K f is well defined: Let Kg = Kg 2, g1,g 2 1 ∈ G Then kg , k ∈ ker ( f ) K, g1 2 = = and ( g ) = f ( kg ) = f ( k ) f ( g ) = ef ( g ) f ( ) f = 1 2 2 2 g2 is a homomorphism since ( Kg Kg ) = f ( Kg g ) = f ( g g ) f ( g ) f ( ) f = 1 2 1 2 1 2 1 g 2 (Θ f is a homomorphism).
Page 1 and 2: 1 Advanced Abstract Algebra M.A./M.
Page 3 and 4: Contents 3 Unit I 5 Unit II 49 Unit
Page 5 and 6: UNIT-I 5 Unit-I Definition Group A
Page 7 and 8: UNIT-I 7 2. In example 1, a 1 1 0 1
Page 9 and 10: UNIT-I 9 Solutions: 1. CGbg≠ a φ
Page 11 and 12: UNIT-I 11 ∴ Ha ⊆ H. To show let
Page 13 and 14: UNIT-I 13 Corollary 4: (Feremat's L
Page 15: UNIT-I 15 Definition: Group Homomor
Page 19 and 20: UNIT-I 19 ( G ) N G ( H ) ≅ H N D
Page 21 and 22: UNIT-1 21 Theorem 8. Let H be a sub
Page 23 and 24: UNIT-1 23 From (i) and (ii), we hav
Page 25 and 26: UNIT-I 25 1 Note that αβ = 2 2 1
Page 27 and 28: UNIT-1 27 In this case the length o
Page 29 and 30: UNIT-1 29 | n Theorem 18. The set o
Page 31 and 32: UNIT-I b = φ ( φ φ ) ( ) g h t x
Page 33 and 34: UNIT-I 33 This is impossible. So no
Page 35 and 36: UNIT-I 35 integers x and y. Note th
Page 37 and 38: UNIT-I 37 O(7) = 4, as 7 1 = 7, 7 2
Page 39 and 40: UNIT-I 39 Put a x mt , = b = x Then
Page 41 and 42: UNIT-I n iii. N = ∈Gs, show that
Page 43 and 44: UNIT-I 43 abab------ab = a n b n
Page 45 and 46: UNIT-I Q.20. Q.21. Q.22. Q.23. Q.24
Page 47 and 48: UNIT-I 47 Remark : If G is abelian,
Page 49 and 50: UNIT-IV 81 Definition. Let S be a s
Page 51 and 52: UNIT-IV 83 (r+s) − (r 1 +s 1 ) =
Page 53 and 54: UNIT-IV 85 and = ψ(r+K) + ψ(s+K)
Page 55 and 56: UNIT-IV 87 M ⊂ M′ ⊂ R M′ =
Page 57 and 58: UNIT-IV 89 We define addition and m
Page 59 and 60: UNIT-IV 91 be the set of the ordere
Page 61 and 62: UNIT-IV 93 (iii) a + b − a b
Page 63 and 64: UNIT-IV 95 and Z/A ~ Im(f) = f (z)
Page 65 and 66: UNIT-IV 97 If a n = 0 for all n ≥
Page 67 and 68:
UNIT-IV 99 x 3 = (0, 0, 0, 1,…) .
Page 69 and 70:
UNIT-IV 101 Then either (i) or n i
Page 71 and 72:
UNIT-IV 103 deg f(x) = 0 and deg g(
Page 73 and 74:
UNIT-IV 105 If x, y ∈ A , then th
Page 75 and 76:
UNIT-IV 107 The possibility φ(r) <
Page 77 and 78:
UNIT-IV 109 Lemma 4. If f(x) is an
Page 79 and 80:
UNIT-IV 111 Proof. Let, if possible
Page 81 and 82:
UNIT-II 49 Unit-II Definition: Comp
Page 83 and 84:
UNIT-II 51 If r = 1, then G is simp
Page 85 and 86:
UNIT-II 53 group of the Commutator
Page 87 and 88:
UNIT-II di + 1 idi + 1 i di + 1 idi
Page 89 and 90:
UNIT-II 57 we write the refinement
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UNIT-II 59 of length e in which eac
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UNIT-II 61 of nilpotency of G) , so
Page 95 and 96:
UNIT-III 63 n.v = R i.e. abelian gr
Page 97 and 98:
UNIT-III 65 Theorem 1 Let M, N be R
Page 99 and 100:
UNIT-III 67 F ( BA ad 1'= = , 1a d(
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UNIT-III 69 Vector space over F. Le
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UNIT-III 71 1. J = only one linearl
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UNIT-III Hence J = −L NM 4 0 0 0
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UNIT-III 75 Example 5 The direct su
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UNIT-IV 77 A. B ≠ B. A. If IR, th
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UNIT-IV 79 Example: (D, +, .) is a
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UNIT-IV 81 Examples 1. The polynomi
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UNIT-V 129 (Characteristic of a rin
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UNIT-V 131 As F = p dΘ F ≅ Z p i
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UNIT-V 133 In this case is called t
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+ c c+ 0 0 1 c c+ 1 1 1 0 1+ c c c
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UNIT-V 137 element of defines a per
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Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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