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

Paper code: 13511
B.Sc. (Computer Science) (Part 2)
Examination, 2016
Paper No. 1.2

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.


1. (a) Show that the set of matrices:

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

Where \alphais 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


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.


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.


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.


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} .


Lokesh Kumar

Being EASTER SCIENCE's founder, Lokesh Kumar wants to share his knowledge and ideas. His motive is "We assist you to choose the best", He believes in different thinking.

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