Loyola College B.Sc. Mathematics April 2012 Modern Algebra Question Paper PDF Download








Date : 21-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00



ANSWER ALL THE QUESTIONS:                                                                     (10 x 2 = 20 marks)


  1. Define an equivalence relation.
  2. Show that if every element of the group G is its own inverse then G is abelian.
  3. Show that every cyclic group is abelian.
  4. Define a normal subgroup of a group
  5. Define the kernel of a homomorphism of a group.
  6. Express (1,3,5) (5,4,3,2) (5,6,7,8) as a product of disjoint cycles.
  7. If A is an ideal of a ring R with unity and 1A show that A=R.
  8. If F is a field show that its only ideals are and F itself.
  9. Define a maximal ideal.
  10. Show that every field is a principal ideal domain.




ANSWER ANY FIVE QUESTIONS:                                                                    (5 x 8 = 40 marks)


  1. Show that a non empty subset H of a group G is a subgroup of G if and only if a,bH implies that ab-1
  2. Show that every subgroup of a cyclic group is cyclic.
  3. State and prove Langrange’s theorem.
  4. Show that a subgroup N of a group G is a normal subgroup of G if and only if every left coset of N in G is a right coset of N in G.
  5. State and prove Cayley’s theorem.
  6. Show that every finite integral domain is a field.
  7. Show that every Euclidean ring is a principal ideal domain.
  8. Let R be a Euclidean ring. Show that any two elements a and b in R have a greatest common divisor d which can be expressed in the form d=λa+mb for l, m in R.




ANSWER ANY TWO QUESTIONS:                                                                    (2 X 20 = 40 Marks)


  1. (i) If H and K are finite subgroup of a group G, show that

(ii) Show that a group G cannot be the union of two proper subgroups.                               (12+8)



  1. (i) If G is a group and N is a normal subgroup of G, show that G/N, the set of all distinct left

cosets of N in G is also a group.


(ii) If H is the only subgroup of order o (H) in the group G, show that H is normal in G.       (12+8)


  1. (i) If H and N are subgroups of a group G and suppose that N is normal in G, show that is

isomorphic to .

(ii) If R is a commutative ring with unit element whose only ideals are (O) and R itself, show that

R is a field.                                                                                                                        (10+10)


  1. (i) Show that an ideal of the Euclidean ring R is a maximal ideal if and only if it is generated by a

prime element of R.


(ii) Show that the characteristic of an integral is either zero or a prime number.                    (14+6)



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