# Abstract Algebra 2016 – BSc Computer Science Part 2 (MJPRU)

#### Paper code: 13511 1511 B.Sc. (Computer Science) (Part 2) Examination, 2016 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 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) Show that the set of all positive rational numbers form an abelian group under the composition defined by:

$a*b= \frac{ab}{2}$

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

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

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

Section-B

3. (a) State and prove Lagrange’s theorem.

(b) Use Fermat’s theorem to determine the remainder, if $8^{103}$ is divided by 103.

4. (a) Show the every subgroup of a cyclic group is cyclic.

(b) Prove that the intersection of any two normal subgroups of a group is a normal subgroup.

Section-C

5. (a) Show that the set R={0,1,2,3,4,5} is a commutative ring with respect to $'+_{6}'$ and $'\times_{6}'$ as two ring compositions.

(b) Prove that ever field is an internal domain.

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

(b) State and prove Fundamental theorem on homomorphism of rings.

Ssection-D

7. (a) Show that S is an ideal of S+T, where S is any ideal of ring R, and T any sub-ring of R.

(b) Prove that the necessary and sufficient condition that the non-zero element a in the Euclidean ring R is a unit is that d(a)=d(1).

8. Define maximal ideal and shown that an ideal S of a ring R with unity is maximal if an only if the residue class ring R/S is a field.

Section-E

9. (a) Show that the union of two sub-spaces is a subspace if and only if one is contained in the other.

(b) If $\alpha_{1}$ and $\alpha_{2}$ are two vector V(F), and $a,b\in F$ , show that the set $\begin{Bmatrix}\alpha_{1},\alpha_{2}, a\alpha_{1} +b\alpha_{2}\end{Bmatrix}$ is linearly dependent.

10. If $W_{1}$ and $W{1}$ are two sub-spaces of a finite dimensional vector space V(F), then: $dim \begin{pmatrix}W_{1}+W_{2}\end{pmatrix}= dim W_{2}+ dim \begin{pmatrix}W_{1} \cap W_{2}\end{pmatrix}$ .

……..End……..

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