Abstract Algebra 2019 – BSc Computer Science Part 2 (MJPRU)

Paper code: 13511
1511
B.Sc. (Computer Science) (Part 2)
Examination-2019
Paper No. 1.2
ABSTRACT ALGEBRA

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.

Section-A

1. (a) Show that the set of all positive relation numbers forms an abeliangroup under the composition defined by :

    (b) Prove that the set is a finte abelian group of order 6 with respect to multiplication modulo 7.

2. (a) Define a permutation. If and

    (b) Show that the intersection of any family of subgroups of a group is a subgroup.

3. (a) State and prove Lagrange’s theorem.

    (b) Show that the intersection of any two normal subgroups of a group is a normal subgroup.

Section-B

4. If H is a normal subgroup of a group G and K a normal subgroup of G containing H, then

5. (a) Show that S is an ideal of S+T where S is my ideal of ring R and T any subring of R.

    (b) Prove that. If a is an element in a commutative ring R with unity, then the set is a principal ideal of R generated by the element a.

6. (a) Show that every homomorphic image of a ring R is isomorphic to some residue class ring thereof.

    (b) Show that the ring of integer is Euclidean ring.

Section-C

7. (a) Show that the linear span L(S) of any subset S of a vector space V(F) is a sub-space of V generated by S.

    (b) Show that is V(F) is a finite dimensional vector space, then any two basis of V have the same number of elements.

8. If are two subspaces of finite dimensional vector space V(F), then

9. (a) State and prove isomorphism theorem for vector space.

    (b) Show that the vector form a basis of :

……..End……..

Thank You!