# Abstract Algebra 2018 – BSc Computer Science Part 2 (MJPRU)

#### Paper code: 135111511B.Sc. (Computer Science) (Part 2)Examination, 2018Paper No. 1.2ABSTRACT 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 matrices:

$A_{\alpha} = \begin{bmatrix} \cos \alpha &-\sin \alpha \\\sin \alpha & \cos \alpha \end{bmatrix}$

Where $\alpha$is a real number, forms a group under matrix multiplication?

(b) Prove that the set of all n nth roots of unity forms a finite abelian group of order n with respect to multiplication.

2. (a) Show that every permutation can be expressed as a product of disjoint cycles.

(b) Show that a necessary and sufficient condition for a non-empty subset H of a group G to be a subgroup is that :

$a,b \in H \Rightarrow ab^{-1} \in H$

where $b^{-1}$ is inverse of b in G.

3. (a) Show that order of each subgroup of a finite group is a divisor of the order of the group.

(b) Show that every group of prime order is cyclic.

Section-B

4. (a) Show that intersection of any two normal subgroup of a group is a normal subgroup.

(b) State and prove fundamental theorem on homomorphism of groups.

5. (a) Show that ever field is an integral domain.

(b) Show that a ring R is withput zero divisors if and only if the cancellation laws holds in R.

6. (a) Show that intersection of two subrings of a ring R is also a subring of R.

(b) Show that S is an ideal of S+T, where S is any ideal of ring R and T an subring of R.

Ssection-C

7. (a) If f is a homomorphism of a ring R into a ring R’ with kernel S, then S is an ideal or R.

(b) An ideal S of the ring of integers I is maximal iff S is generated by some prime integer.

8. (a) Show that union f two subspaces is a subspace if and only if one is contained in the other.

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

9. (a) Show that the three vectors (1, 1, -1), (2,-3, 5) and (-2, 1, 4) of R3 are linearly independent.

(b) if W be a subspace of a finite dimensional vector-space, then show that :

$\dim \frac{V}{W} = \dim V - \dim W$

10. Discuss the direct sum of subspaces and show that the necessary and sufficient conditions for a vector space V(F) to be a direct sum of its two subspaces W1 and W2 are that :

(i) $V = W_{1} + W_{2}$

(ii) $W_{1} \cap W_{2} = \left \{ \bar{O} \right \}$

……..End……..

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