Abstract Algebra 2023 – 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-2023
Paper No. 1.2
ABSTRACT ALGEBRA

Time: Three Hours] [Maximum Marks: 50

Note: Attempt five questions in all selection at least one question from each section. All questions carry equal marks.

Section-A

1. (a) Prove that the set of all cosets of a normal subgroup is a group with respect to multiplication of complexes as composition.

    (b) If H is a normal subgroup of a solvable group G, then the quotient group G/H is solvable.

2. (a) State and prove Cayley’s theorem.

    (b) Prove that every subgroup of a cyclic group in cyclic.

3. (a) State and prove fundamental theorem of homomorphism of groups.

    (b) Show that every permutation can be ex-pressed as product of disjoint cycles.

Section-B

4. (a) Show that the set of matrices  is a subring of the ring of 2 x 2 matrices with integral elements.

    (b) Prove that the ring of integers is a principle ideal ring.

5. (a) Prove that a field has no proper ideals.

    (b) Show that every finite integral domain is a field.

6. (a) State and prove unique factorization theorem.

    (b) Show that the set R={0, 1, 2, 3, 4, 5} is commutative ring with respect to ‘+₆‘ and ‘X₆‘ as two ring compositions.

7. (a) Prove that if two vectors are linearly dependent one of them is a scalar multiple of the other.

    (b) Prove that the necessary and sufficient condition that the non-zero element a in the Euclidean ring R is a unit is that d(a) = d(1).

8. (a) Show that the three vectors (1, 1, -1), (2, -3, 5) and (-2, 1, 4) of R³ are linearly independent.

    (b) If W be a subspace of a finite dimensional vector space, then show that :

9. (a) Define basis of a vector space and show that there exists a basis for each finite dimensional vector space.

    (b) Define direct sum of two subspaces and show that the necessary and sufficient conditions for a vector space V(F) to be a direct sum of its two subspaces W₁ and W₂ are that-

         1. v = w₁ + w₂

         2. w₁ ∩ w₂ = {0}

……..End……..

Thank you 🙂