Abstract Algebra 2021 – BSc Computer Science Part 2

Total No. of Questions : 9] [Total No. of Printed Pages: 4

 

Paper code: 13511
1511
B.Sc. (Computer Science) (Part 2)
Examination-2021
Paper No. 1.2
ABSTRACT ALGEBRA

Time: Hours] [Maximum Marks: 50

 

Section-A

Note: Attempt any two questions.15 each

1. (a) If G is a group then prove that:

    (b) Show that the set of matrices

     where is a real number, forms a group under matrix multiplication.

2. (a) The set Pn of all permutations on n symbols is a finite group of order n! with respect to composite of mapping as the operation.

    (b) Show that the union of two subgroups is a subgroup if and only if one is contained in the other.

3. (a) Show that any two right cosets of a subgroup are either disjoint or identical.

    (b) State and prove Cayley’s theorem.

Section-B

Note: Attempt any one questions.20 each

4. (a) A subgroup H of a group G is normal if and only if

    (b) if N and M are normal subgroups of G, prove that NM is also a normal subgroup of G.

5. (a) A ring R is without zero divisor if and only if the cancellation laws hold in R.

    (b) Show that every field is an integral domain.

6. (a) A commutative ring with unity is a field if it has no proper ideal.

    (b) The ring of integers is a principal ideal ring.

7. (a) Show that the set W of the elements of the vector space V₃(R) of the form , where is a subspace of V₃(R).

    (b) Show that the vectors (1, 1, 2, 4), (2, -1, -5, 2), (1, -1, -4, 0) and (2, 1, 1, 6) are lineraly independent in R⁴.

8. Each subspace W of a finite dimensional vector space V(F) of dimension n is a finite dimensional space with dim m ≤ n. Also V = W iff dim V = dim W. 

9. (a) Every n dimensional vector space V(F) is isomorphic to V_n(F).

    (b) If a finite dimensional vector space V(F) is a direct sum of two subspaces w₁ and w₂ then

dim V = dim w₁ + dim w₂

……..End……..

Thank You 🙂