Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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60 ADVANCED ABSTRACT ALGEBRA G15 G1 ≅ G5 G1 under the mapping G 2 a G 4 a G ↔ G 5 1 5 5 10 ↔ a G 1 20 ↔ a G 1 G5 G1 ≅ G3 G6 under the mapping G 6 a G a G a G a G ↔ G 1 6 ↔ aG 1 6 12 2 1 ↔ a G6 18 3 1 ↔ a G6 24 4 1 ↔ a G6 Multiplication table for factor groups for G 2 4 G a G a G 5 2 4 G G a G a G 5 5 2 2 4 a G a G a G G 5 5 4 4 2 a G a G G a G 5 5 5 5 5 5 5 5 5 5 G 15 5 The multiplication table for G G 3 1 10 20 G a G a G 1 1 1 10 20 G G a G a G 1 1 1 1 10 10 20 1 1 1 1 20 20 10 1 1 1 1 a G a G a G G a G a G G a G The isomorphism of G G and G G 15 5 3 can be easily seen from above tables. 1 Remark : Above is very good example of the Jordan-Holder Theorem. Example 12 : Any nilpotent group is solvable. Solution : By the definition of the k th center, each Zkbg G Zkbgis G abelin, so any commutator of two elements of Z k (G) must lie in Z k-1 (G) (see (iv) of example 12 of section I). Hence, if G is nilpotent of class c, (A finite group G is defined to be nilpotent when there is some index c with Z c (G)=G, the first such index c is called the class
UNIT-II 61 of nilpotency of G) , so Z c (G) = G. Now G'( . So we get the derived series . Hence G is solvable. Note that the converse is not true i.e. a solvable group need not be nilpotent. For example : G ∆A ∆bg 3 3 1 S 3 is solvable but is not nilpolent because center of S 3 is (1). (Every nilpotent group has a non-trivial center) Exmple 13 : Consider b g bg b g bg . Now Z Z × S = Z × 1 and every Z × S = Z × 1 . Therefore, ascending 1 2 3 2 Z k 2 3 2 central series (or upper central series) never reach G. Hence Z × S is not niloptent. 2 3 ( ) l q l1q c Zc−G1 , GG' cZe 2 , − G, c −1 G = ∆ Z b g ( ' ∆ ' ∆ ∆ ∆= 1 c ) 2 × S3 − − − G =
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1 Advanced Abstract Algebra M.A./M.
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Contents 3 Unit I 5 Unit II 49 Unit
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UNIT-I 5 Unit-I Definition Group A
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UNIT-I 7 2. In example 1, a 1 1 0 1
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UNIT-I 9 Solutions: 1. CGbg≠ a φ
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UNIT-I 11 ∴ Ha ⊆ H. To show let
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UNIT-I 13 Corollary 4: (Feremat's L
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UNIT-I 15 Definition: Group Homomor
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UNIT-I 17 Proof: Consider the diagr
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UNIT-I 19 ( G ) N G ( H ) ≅ H N D
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UNIT-1 21 Theorem 8. Let H be a sub
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UNIT-1 23 From (i) and (ii), we hav
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UNIT-I 25 1 Note that αβ = 2 2 1
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UNIT-1 27 In this case the length o
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UNIT-1 29 | n Theorem 18. The set o
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UNIT-I b = φ ( φ φ ) ( ) g h t x
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UNIT-I 33 This is impossible. So no
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UNIT-I 35 integers x and y. Note th
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UNIT-I 37 O(7) = 4, as 7 1 = 7, 7 2
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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
Page 91: UNIT-II 59 of length e in which eac
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(
Page 101 and 102: UNIT-III 69 Vector space over F. Le
Page 103 and 104: UNIT-III 71 1. J = only one linearl
Page 105 and 106: UNIT-III Hence J = −L NM 4 0 0 0
Page 107 and 108: UNIT-III 75 Example 5 The direct su
Page 109 and 110: UNIT-IV 77 A. B ≠ B. A. If IR, th
Page 111 and 112: UNIT-IV 79 Example: (D, +, .) is a
Page 113 and 114: UNIT-IV 81 Examples 1. The polynomi
Page 115 and 116: UNIT-V 129 (Characteristic of a rin
Page 117 and 118: UNIT-V 131 As F = p dΘ F ≅ Z p i
Page 119 and 120: UNIT-V 133 In this case is called t
Page 121 and 122: + c c+ 0 0 1 c c+ 1 1 1 0 1+ c c c
Page 123 and 124: UNIT-V 137 element of defines a per
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Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

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