Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

More documents

Recommendations

Info

8 ADVANCED ABSTRACT ALGEBRA Problem 1: Let G be a non empty set closed under an associative product, which has left indentity e and left inverse for all elements of g. show that G is a group. Proof: Let a ∈G and let b such that b a = e. Now b a b = (b a) b = e b = b ...................(i) such that c b = e Hence c (b a b) = cb = e from (i) bgb g cb a b = e ab = e ∴ b is also right inverse of a. Further, a e = a (b a) = (a b) a = e a = a Hence e is right identity also Thus G is a group, Subgroups Let H be a non-empty subset of the group G such that 1. Θ ∈a ≠∃ 2. a − 1 ∈H ∀a ∈H We prove that H is a group with the same law of composition as in G. Proof: H is closed under multiplication from (1). All elements of H are from G and associative law holds in G, therefore, multiplication is associative in H also. Let a ∈ H , then a -1 from (2) and so from (1), a a -1 , i.e. e = a a -1 . which implies, identity law holds in H, (2) gives inverse law in H. Thus H is a group. H is called a subgroup of G. Thus a nonempty subset of a group G which is a group under the same law of composition is called a subgroup G. Note that e, the identity element G is also the identity of H. A group G is called nontrivial if G (e). A nontrivial group has at teast two subgroups namely G and (e). Any other subgroup is called a proper subgroup. Definition: Let b, a Problems: G, b is said to be Conjugate of a 1. Let a ∈G, let C G (a) = {x 2. G: x -1 ax = a} Prove that C G (a) is a sbgroup of G. G, if is a sub group of G. such that b = x -1 ax. 3. Find the centre of the group GL (2, R) of nonsingular 2 x 2 matrics over real numbers, Zb
UNIT-I 9 Solutions: 1. CGbg≠ a φ,because e∈C G (a). Let x,y C G (a). Then bg bbg bg g bg 1 d ∈− 1 1 Zx xy xy , y G, ax x∈ − C= ∈ −1 −1 Ga ΙxC ⇔ Za yG C ax a= Thus a= , ∀y xa a∈∀ Gx xy, a, ∈hence Gx a -1 , x∈C y G (a), hence C G (a) is subgroup of G a∈G G Remarks: −1 −1 = y dx axiy −1 so C = y a ycΘ x∈C Gbg a h G (a) is the set of all elements of G commuting with a. we call C = a cΘ y∈ G (a), the centralizer of a CGbg a h xy ∈C a 2. . Let x, y ∈Zbg G . From above G bg Also, x a x = x a x x ∈C a i b g −1 −1 −1 −1 d i d i −1 G bg = x x ax x Θ x ∈CG a −1 −1 = dx x i adx x i = e a e = a d i −1 −1 c bgh hence Note that Thus Z (G) is a subgroup. Definition Z (G) is called the center of the group G. 3. let x a c b = F H G I K J ∈ centre of GL (2, IR) d ∴ x commutes with all non-singular 2 x 2 matrices, So in particular x commutes with
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: UNIT-I 7 2. In example 1, a 1 1 0 1
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 and 16: UNIT-I 15 Definition: Group Homomor
Page 17 and 18: UNIT-I 17 Proof: Consider the diagr
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
Page 91 and 92:
UNIT-II 59 of length e in which eac
Page 93 and 94:
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(
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
show all

Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?