Abstract Algebra 2017 – BSc Computer Science Part 2 (MJPRU)

Paper code: 13511
1511
B.Sc. (Computer Science) (Part 2)
Examination, 2017
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 I of all integers is an abelian group with respect to the operation * defined by:

(b) Show that the set  forms an abelian group with respect to multiplication.

2. (a) What do you know about even and odd permutations and prove that out of  permutations  are even permutations and  are odd permutations.
(b) Show that the intersection of two subgroup of a group G, is also a subgroup of G.

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

(b) Show that a subgroup H of a group G is normal if and only if .

Section-B

4. (a) Show that every homomorphic image of a group G is isomorphic to some quotient group of G.

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

5. (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.

6. (a) Show that if D is an integral domain, then the polynomial ring  is also an integral domain.

(b) State and prove unique factorization theorem.

Section-C

7. (a) Show that every Euclidean ring is a principle ideal ring.

(b) Show 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  and .

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

(b) Is the vector  in the subspace of spanned by the vectors ,  and ?

9. (a) Determine whether or not the following vectors from a basis of :

, , 

(b) If f is a homomorphism of U(F) into V(F), then prove that:

1. where \bar{0} and \bar{{0}’} are the zero vectors of U and V respectively.
2. 

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