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
2. (a) What do you know about even and odd permutations and prove that out of
(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
(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
(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
8. (a) Prove that if two vectors are linearly dependent, one of them is a scalar multiple of the other.
(b) Is the vector
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:
where \bar{0} and \bar{{0}’} are the zero vectors of U and V respectively.
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