Paper code: 13511
B.Sc. (Computer Science) (Part 2)
Paper No. 1.2
Time: Three Hours] [Maximum Marks: 50
Note: Attempt five questions in all selecting at least one question from each Section. All questions carry equal marks.
1. (a) Show that the set of matrices:
Where is a real number, forms a group under matrix multiplication?
(b) Prove that the set of all n nth roots of unity forms a finite abelian group of order n with respect to multiplication.
2. (a) Show that every permutation can be expressed as a product of disjoint cycles.
(b) Show that a necessary and sufficient condition for a non-empty subset H of a group G to be a subgroup is that :
where is inverse of b in G.
3. (a) Show that order of each subgroup of a finite group is a divisor of the order of the group.
(b) Show that every group of prime order is cyclic.
4. (a) Show that intersection of any two normal subgroup of a group is a normal subgroup.
(b) State and prove fundamental theorem on homomorphism of groups.
5. (a) Show that ever field is an integral domain.
(b) Show that a ring R is withput zero divisors if and only if the cancellation laws holds in R.
6. (a) Show that intersection of two subrings of a ring R is also a subring of R.
(b) Show that S is an ideal of S+T, where S is any ideal of ring R and T an subring of R.
7. (a) If f is a homomorphism of a ring R into a ring R’ with kernel S, then S is an ideal or R.
(b) An ideal S of the ring of integers I is maximal iff S is generated by some prime integer.
8. (a) Show that union f two subspaces is a subspace if and only if one is contained in the other.
(b) Prove that if two vectors are linearly dependent, one of them is a scalar multiple of the other.
9. (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 :
10. Discuss the direct sum of 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 :