Download OU B.Sc Computer Science 4th Sem Algebra Important Questions

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Subject Title: Algebra Prepared by: S Shravani

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Year: II Semester: IV Updated on: 23.03.

Unit - I

1. A group G, identity element is unique.
2. In a group G, inverse element is unique.
3. Prove that the set Z of all integers form an abelian group w.r.t the operations defined by a*b = a+b+2, for all a,b ? Z

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4. Cancellation laws holds in a group. (let G be a group. Then for a,b, c ? G, ab=ac ? b = c and ba=ca? b =c.
5. In a group G for a,b,x,y ? G the equation ax = b and ya = b have unique solutions.
6. If every element of a group (G,.) is its own inverse, show that (G,.) is abelian group.
7. The order of every element of a finite group is finite and less than or equal to the order of a group.
8. In a group G if a ? G, then |a|= |a^-1|.

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9. If a is an element of group G such that |a| = n, then a^m=e iff n/m.
10. If a is an element of group G such that |a| = 7, then show that a is the cube of some element of G.
11. A non-empty complex H of a group G is subgroup pf G iff (i) a ? H,b ? H?ab ? H
12. (i) a ? H a^-1 ? H.
13. A non-empty complex H of a group G is subgroup pf G iffa ? H,b ? H ? ab^-1 ? H

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14. The necessary and sufficient condition for a finite complex H of a group G to be a subgroup of G is a ? H,b ? H? ab?H.
15. If H and K are two subgroups of a group G, then HK is subgroup of G iff HK = KH
16. If H₁ and H₂ are two subgroups of a group G, then H₁ n H₂ is also a subgroup of G.
17. The union of two subgroups of a group G is a subgroup iff one is contained in other.
18. Every cyclic abelian group is an abelian group. Converse is not true.

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19. Every subgroup of cyclic group is cylic.
20. If a cyclic group G is generated by an element of order n, the a^m is generator of G iff (m,n) = 1.
21. The order of a cyclic group is equal to its generator.
22. Show that the group ( G= {1,2,3,4,5,6} ,x₇ ) is cyclic . also write down all its generators.
23. How many subgroups does Z₁₂ have? List them.

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24. If G is an infinite cyclic group, the G has exactly two generators which are inverse of each other.
25. Write down the following products as disjoint cycles.
    1. (132)(567)(261)(45)
    2. (i)(136)(1357)(67)(1234).
26. Definitions:
    1. Group

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    2. Subgroup
    3. Addition modulo
    4. Multiplication modulo
    5. Cyclic group
    6. Permutation group

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    7. Order of a group
    8. Order of an element

Unit - II

26. If R is a Boolean ring then (i) a+a=0 for all a ? R (ii) a+b = 0 ? a=b and (iii) R is commutative under multiplication.

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27. Find all units of Z₁₂
28. The intersection of two subring of a ring R is a subring of R, A ring has no zero divisors iff the cancellation laws holds in R, A field has no zero-divisors.
29. List all zero-divisors in Z₁₂, Can you see a relationship between the zero-divisors of Z₁₂ and the units of Z₁₂ ?, A field is an integral domain.
30. Every finite integral domain is a field.
31. The characteristic of an integral domain is either a prime or zero.

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32. A field has no proper non-trivial ideals.
33. A commutative ring R with unity element is a field if Rthave no proper ideals.

Unit IV

34. The union of two ideals of a Ring R is a ideal iff one is contained in other.
35. Every ideal of z is a principal ideal.

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36. An ideal U of a commutative ring R with unity is maximal iff the quotient ring R/U is a field.

37. Find all maximal ideals in a. Z₈. b. Z₄₀. c. Z₁₅.d. Z₇.

38. Definitions:
    1. Ring, Boolean ring, Sub ring
    2. Integral domain
    3. Field

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    4. Zero divisor
    5. Characteristic of a ring
    6. Idempotent and nilopotent
    7. Ideal
    8. Principal ideal

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    9. Maximal ideal
    10. Factor ring
    11. Prime ideals

39. Test by divisibility by 9.

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40. Prove that a ring homomorphism carries an idempotent to an idempotent.
41. The homomorphic image of a ring is a ring.
42. Let R, R' be two rings and f : R ? R' be a homomorphism. For every ideal U'ina ring R' f^-1(U')is an ideal in R.
43. Let f(x) = 4x⁶ + 2x⁵ + x + 3 and g(x) = 3x⁴ + 3x³+ 3x² + x + 4, where f(x), g(x) ? Z₅[x]. Compute f(x) + g(x) and f(x).g(x).
44. If f is a homomorphism of a ring R into a ring R':the ker f is an ideal of R.

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45. If f is a homomorphism of a ring R into a ring R' then R then f is an into isomophism iff
46. Every quotient ring of a ring is a homomorphic image of a ring.

Unit III

40. Fundamental theorem of homomorphism
41. The division algorithm.

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42. Factor theorem.
43. Determine all ring homomorphisms from Z to Z.
44. Definitions:
    1. Homomorphism ring
    2. Isomorphism ring
    3. Homomorphic image of ring

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    4. Monomorphism ring
    5. Automorphism ring
    6. Kernel of a homomorphism of ring
    7. Polynomial ring
    8. Degree of polynomial

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45. Any infinite cylic group is isomorphic to integers Z.
46. Cayley’s theorem.
47. Find the regular permutations group isomorphic to the multiplicative group {1, -1, i, -i}
48. Let (G, *), (G', .) be two groups. let f be a homomorphism from G into G' then (i) f(e) =e' where e is the identity in G and e'is identity in G' . (ii) f(a^-1 )= {f(a)}^-1.

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49. Every homomorphic image of an abelian group is abelian.
50. let f be a isomorphism from G onto G' then G = (a)iff G' = (f(a)) .
51. let f be a isomorphism from G onto G' then for any elements a and b in G, a and b commute iff f(a) and f(b) commute.
52. show that the mapping f : G ? G such that f(a) = a^-1 for all a? G, is an automorphismof a group G iff G is abelian.
53. The set of all automorphism of a group, G forms a group w.r.t composition of mapping.

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54. The set of all inner automorphism of a group G forms a group w.r.t composition of mapping.
55. H is any subgroup of a group (G, *)-and h ? G then h ? H, iff hH=H=Hh.
56. If a and b are any two elements of group G and H is subgroup of group G then
57. Any two left (right) cosets of a subgroup are either disjoint or identical.
58. If H is a subgroup of a group G for a,b ? G the relation a = b(modH) is an equivalence relation.

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59. If H is a subgroup of a group G then there is one-one correspondence between the set of all distinct left cosets of H'in G and the set of all distinct right cosets of H in G.
60. Lagrange’s theorem.
61. If G is a finite group and a ? G, then|a|/|G|.
62. If p is a prime number then every group of order p is cyclic group i.e a group of prime order is cyclic.
63. Orbit — Stabilizer theorem.

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64. A subgroup of H of a group G is normal , if xHx^-1 = H for all x ? G.
65. A subgroup of H of a group G is a normal subgroup of G iff each left coset of H in G is a right coset of H in G.
66. The set A_n of all even permutations on n symbols is a normal subgroup of the permutation group S_n on the n symbols.
67. A subgroup of H of a group G is a normal subgroup of G iff product of two left(right) cosets of H in G is again left(right) coset of H in G.
68. Every subgroup of an abelian group is normal.

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69. If G is a group and H is a subgroup of index 2 in G, then H is normal subgroup of G.
70. The union of two normal subgroups of a group G is a normal subgroup.
71. H is normal subgroup of G. the set © of all cosets of H in G w.r.t coset multiplication is a group

51. Definitions:
    1. Homomorphism Group
    2. Isomorphism Group

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    3. Homomorphic image of a Group
    4. Monomorphism Group
    5. Automorphism Group
    6. Coset
    7. Congruence modulo

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    8. Index
    9. Normal subgroup
    10. Factor group
    11. Kernel of a homomorphism

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