# Abstract Algebra 2015 – BSc Computer Science Part 2 (MJPRU)

#### Paper code: 13511 1511 B.Sc. (Computer Science) (Part 2) Examination, 2015 Paper No. 1.2 ABSTRACT ALGEBRA

##### Time: Three Hours] [Maximum Marks: 50

Note: Attempt five questions in all section one question from each Section. All questions carry equal marks.

Section-A

1. (a) Show that the set containing four fourth root of unity from a group with respect to multiplication.

(b) Show that out of Permutations on n! symbols,  n/2! are even permutation and n/2! are odd permutations.

2. (a) Show that the intersection of two subgroups of a group G, is again a subgroup of G.

(b) Define costs with example by giving index of a subgroup.

or

Prove that every subgroup of a cyclic group in cyclic.

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

(b) Show that A subgroup of a group G is normal if and only if:

$xHx^{-1}=H\forall x=G$

4. (a) State and prove fundamental theorem of homomorphism of groups.

(b) If H is a subgroup of G and N is a normal of G, show that N is a normal subgroup of HN.

Section-B

5. (a) Show that a finite commutative ring without zero divisors is a field.

(b) Show that a necessary and sufficient conditions for a non-empty subset S of a ring R to be a subring of R are:

1. $a,b \in S \Rightarrow a-b \in S$
2. $a,b \in S \Rightarrow a.b \in S$

6. (a) Prove that a field has no proper ideals.

(b) 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.

7. Show that an ideal S of the ring of integers I is maximal iff S is generated by some prime integer.

Section-C

8. (a) Show that every Euclidean ring is a principal ideal ring.

(b) Prove 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 $a,b \in F, \alpha, \beta \in W \Rightarrow a\alpha +b\beta \in W$ .

9. (a) Prove that of two vector are linearly dependent, one of them is a scalar multiple of the other.

(b) Define basis of a vector space and show that there exists a basis for each finite dimensional vector space.

10. (a) Define direct sum of two subspaces and show that the necessary and sufficient conditions for a vector space V (F) to be a direct sum of its two subspaces $w_{1}$ and $w_{2}$ are that:

1. $\vartheta = w_{1} + w_{2}$
2. $w_{1} \cap w_{2} = \begin{Bmatrix}{0}\end{Bmatrix}$

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

……..End……..

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