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

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

a*b=a+b+1:\forall a,b\in I

(b) Show that the set G={1,-1,i,-i} forms an abelian group with respect to multiplication.

2. (a) What do you know about even and odd permutations and prove that out of n! permutations \frac {n}{2}! are even permutations and \frac{n}{2}! 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 xHx^{-1}=H, \forall x\in G.


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 \begin{bmatrix} a & b\\ 0 & c \end{bmatrix} 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 D\left [ x \right ] is also an integral domain.

(b) State and prove unique factorization theorem.


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 a, b\in F and \alpha ,\beta \in W\Rightarrow a\alpha +b\beta\subset W.

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

(b) Is the vector \left ( 3, -1,0,-1 \right ) in the subspace of R^4 spanned by the vectors \left ( 2, -1,3,2 \right )\left ( -1, 1,1,-3 \right ) and \left ( 3, -1,0,-5 \right )?

9. (a) Determine whether or not the following vectors from a basis of R^3:

\left ( 1,1,2 \right )\left ( 1,2,5 \right )\left ( 5,3,4 \right )

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

  1. f\left ( \bar{0} \right )=\bar{{0}'} where \bar{0} and \bar{{0}’} are the zero vectors of U and V respectively.
  2. f(-\alpha)=-f(\alpha):\forall \alpha \in U


Lokesh Kumar

Being EASTER SCIENCE's founder, Lokesh Kumar wants to share his knowledge and ideas. His motive is "We assist you to choose the best", He believes in different thinking.

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