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IGNOU BMTC-134 - Algebra

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IGNOU BMTC-134 Code Details

  • University IGNOU (Indira Gandhi National Open University)
  • Title Algebra
  • Language(s)
  • Code BMTC-134
  • Subject Mathematics
  • Degree(s) BAG, BSCG
  • Course Core Courses (CC)

IGNOU BMTC-134 English Topics Covered

Block 1 - Introduction to Groups

  • Unit 1 - Some Preliminaries
  • Unit 2 - Groups
  • Unit 3 - Subgroups
  • Unit 4 - Cyclic Groups

Block 2 - Normal Subgroups and Group Homomorphisms

  • Unit 1 - Lagrange’s Theorem
  • Unit 2 - Normal Subgroups
  • Unit 3 - Quotient Groups
  • Unit 4 - Group Homomorphisms
  • Unit 5 - Permutation Groups

Block 3 - Introduction to Rings

  • Unit 1 - Rings
  • Unit 2 - Subrings
  • Unit 3 - Ideals
  • Unit 4 - Ring Homomorphisms

Block 4 - Integral Domains

  • Unit 1 - Integral Domains and Fields
  • Unit 2 - Polynomial Rings
  • Unit 3 - Roots and Factors of Polynomials
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IGNOU BMTC-134 (January 2023 - December 2023) Assignment Questions

IGNOU BMTC-134 (January 2022 - December 2022) Assignment Questions

PART-A 1. Which of the following statements are true? Give reasons for your answers. i) If a group G is isomorphic to one of its proper subgroups, then G = Z. ii) If x and y are elements of a non-abelian group (G, *) such that x * y = y * x, then x=e or y=e, where e is the identity of G with respect to *. iii) There exists a unique non-abelian group of prime order. iv) If v) If H and K are normal subgroups of a group G, then 2. a) Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that (Q+1) is not cyclic. b) Let G be an infinite group such that for any non-trivial subgroup H of Then prove that c) Prove that a cyclic group with only one generator can have at most 2 elements. 3. a) Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic. b) Let be a fixed odd permutation in S10. Show that every odd permutation in S10 is a product of and some permutation in A10. c) List two distinct cosets of <r> in D10, where r is a reflection in D10. d) Give the smallest for which An is non-abelian. Justify your answer. 4. Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma: Let H and K be subgroups of a group G and H’ and K’ be normal subgroups of H and K, respectively. Then The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’. PART-B 5. Which of the following statements are true, and which are false? Give reasons for your answers. i) For any ring R and ii) Every ring has at least two elements. iii) If R is a ring with identity and I is an ideal of R, then the identity of R/ I is the same as the identity of R. iv) If is a ring homomorphism, then it is a group homomorphism from v) If R is a ring, then any ring homomorphism from R x R into R is surjective. 6. a) For an ideal I of a commutative ring R, define b) Is for any two ideals I and J of a ring R? Give reasons for your answer. 7. Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and • on S by: Show that (S, +, •) is a ring isomorphic to R. PART-C 8. Which of the following statements are true, and which are false? Give reasons for your answers. i) If k is a field, then so is k x k. ii) If R is an integral domain and I is an ideal of R then, Char (R) = Char (R/I). iii) In a domain, every prime ideal is a maximal ideal. iv) If R is a ring with zero divisors, and S is a subring of R then, S has zero divisors. v) If R is a ring and is of degree , then f(x) has exactly n roots in R. 9. a) Find all the units of b) Check whether or not is a field. c) Construct a field with 125 elements.
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