2015

(May)

MATHEMATICS

(Major)

Course: 603

(Algebra II and Partial Differential Equations)

Full Marks: 80

Pass Marks: 32

Time: 3 hours

The figures in the margin indicate full marks for the questions.

A: Algebra II

(Marks: 40)

1. (a) Choose the correct answer for the following: 1
Isomorphism of a group onto itself is called

- Automorphisms.
- Homomorphism.
- Endomorphism.
- Ideal.

(b) Suppose a group G is an internal direct product of its subgroups H and K, then write the element which is common to both H. and K. 1

(c) Let G be a group and and automorphisms of G. If , then prove that . 3

(d) Prove that if G is a group, their Aut (G), the set of automorphisms of G is also a group under composition of functions. 4

(e) Let ,be normal subgroups of a group. Then prove that is an inner direct product of and if an only if (i) and (ii) 4

Or

If such that is an automorphisms, where is some fixed integer, then show that for all, is the centre of group

2. (a) Write the binary compositions needed to form a ring in a non-empty set. 1

(b) Write an example of a commutative ring without unity. 1

(c) If R is a ring such that then prove that. 3

(d) Show that a Boolean ring is commutative. 4

(e) Prove that a commutative ring R is an integral domain if and only if for all 4

Or

Show that intersection of any two left ideals of a ring is again a left ideal of the ring.

3. (a) Write the another name of the ring of all equivalent classes. 1

(b) Let be an onto homomorphism, where is a ring with unity. Show that is unity of 3

(c) If be an onto homomorphism, then prove that is isomorphic to a quotient ring of . 5

Or

Prove that any ring can be imbedded into a ring with unity.

(d) Let be a commutative ring with unity. Prove that an ideal of is maximal ideal of if is a field. 5

Or

Show that in a Boolean ring, every prime ideal is maximal.

B: Partial Differential Equations

(Marks: 40)

4. (a) Write the number of arbitrary constants that the solution of the partial differential equation , where possesses. 1

(b) Write the auxiliary equations of the equation. 1

(c) Solve any two equations: 4x2=8

(d) Solve any one equation: 5

(e) Find the integral surface of the linear partial differential equation. 5

which contains the straight line?

5. (a) In Charpit’s method for solving a partial differential equation, a second equation is introduced. Write the order of the equation. 1

(b) Write the complete integral of the equation 1

(c) Write the Charpit’s auxiliary equations for the equation. 2

(d) Find complete integrals of any three of the following: 4x3=12

(e) Find a complete integral of using Jacobi’s method. 4