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 Vn(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 🙂