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

June 19, 2019 Lokesh Kumar 119 Views 0 Comments 2nd Year, BSc, Computer Science, Exam paper

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 :

$a \ast b = \frac{(ab)}{2}$

(b) Prove that the set $G={1,2,3,4,5,6}$ is a finte abelian group of order 6 with respect to multiplication modulo 7.

2. (a) Define a permutation. If $A=\begin{pmatrix}1 & 2 & 3\\2 & 3 & 1\end{pmatrix}$ and $B=\begin{pmatrix}1 & 2 & 3\\3 & 1 & 2\end{pmatrix}$

(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

$G/K \cong (G/H) / (K/H)$

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 $S = \left \{ ra : r\in R \right \}$ 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 $w_{1}, w_{2}$ are two subspaces of finite dimensional vector space V(F), then $\dim \left (w_{1} +w_{2} \right )= \dim w_{1}+\dim w_{2} - \dim (w_{1}\cap w_{2}).$