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

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

Time: Three Hours] [Maximum Marks: 50


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


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.


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.


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.


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.


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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