Loyola College B.Sc. Mathematics April 2008 Modern Algebra Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ  13

 

FOURTH SEMESTER – APRIL 2008

MT 4502 / 4500 – MODERN ALGEBRA

 

 

 

Date : 26/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL questions.:                                                                               (10 x 2 = 20)

  1. Define an equivalence relation on a set.
  2. Define a binary operation on a set.
  3. Define a cyclic group.
  4. Define a quotient group of a group.
  5. Define an isomorphism.
  6. Define a permutation group.
  7. Define a division ring.
  8. Define a field.
  9. Define an integral domain.
  10. What is a Gaussian integer?

 

PART – B

Answer any FIVE  questions.                                                                      (5 x 8 = 40)

  1. If G is a group, then prove that
  • for every
  • for all
  1. Prove that anon – empty subset H of a group G is a subgroup of G if and only if

(i)

(ii)

  1. If H is a subgroup of a group G, then prove that any two left Cosets of H in G either are identical or have no element in common.
  2. If H is a subgroup of index 2 in a group G, prove that H is a normal subgroup.
  3. If  is a homomorphism of a group G into a group , prove that

(i) , the identity element of G1

(ii) for all

  1. Show that the additive group G of integers is isomorphic to the multiplicative group
  2. Prove that the intersection of two subrings of a ring R is a subring of R.
  3. Find all the units in Z(i).

 

PART – C

Answer any TWO   questions.                                                                      (2 x 20 = 40)

  1. State and prove the Fundamental theorem of arithmetic.
  2. a) State and prove Lagrange’s theorem.
  1. b) Show that every subgroup of an abelian group is normal. (14+6)
  1. a) State and prove the fundamental theorem of homomorphism on groups.
  1. b) Define an endomorphism an epimorphism and an automorphism.
  1. State and prove unique factorization theorem.

 

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