Sunday, August 2, 2020

ABSTRACT ALGEBRA - 2017 | B.SC CS 2ND YEAR | MJPRU | EXAM PAPER | My CS Tutorial

My CS Tutorial August 02, 2020

Aabstract algebra | My CS Tutorial

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:

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

(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 $xHx^{-1}=H, \forall x\in G$ .

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 $\begin{bmatrix} a & b\\ 0 & c \end{bmatrix}$ 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.

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 $a, b\in F$ 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:

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

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