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 section one question from each Section. All questions carry equal marks.
1. (a) Show that the set containing four fourth root of unity from a group with respect to multiplication.
(b) Show that out of Permutations on n! symbols, n/2! are even permutation and n/2! are odd permutations.
2. (a) Show that the intersection of two subgroups of a group G, is again a subgroup of G.
(b) Define costs with example by giving index of a subgroup.
Prove that every subgroup of a cyclic group in cyclic.
3. (a) State and prove Cayley’s theorem.
(b) Show that A subgroup of a group G is normal if and only if:
4. (a) State and prove fundamental theorem of homomorphism of groups.
(b) If H is a subgroup of G and N is a normal of G, show that N is a normal subgroup of HN.
5. (a) Show that a finite commutative ring without zero divisors is a field.
(b) Show that a necessary and sufficient conditions for a non-empty subset S of a ring R to be a subring of R are:
6. (a) Prove that a field has no proper ideals.
(b) If f is a homomorphism of a ring R into a ring R’ with kernel S, then show that S is an ideal of R.
7. Show that an ideal S of the ring of integers I is maximal iff S is generated by some prime integer.
8. (a) Show that every Euclidean ring is a principal ideal ring.
(b) Prove that the necessary and sufficient condition for a non-empty subset W of a vector space V (F) to be a subspace of V is .
9. (a) Prove that of two vector are linearly dependent, one of them is a scalar multiple of the other.
(b) Define basis of a vector space and show that there exists a basis for each finite dimensional vector space.
10. (a) 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 and are that:
(b) If and are vectors of V(F) and , show that the set is linearly dependent.