Syllabus

🔹 Unit I – Group Theory (Basics)

Hours: 08 | Marks: 08

1) Definition of Group

Group म्हणजे असा set G ज्यामध्ये operation defined असतो आणि खालील चार properties satisfy होतात:

Group Properties:

1. Closure property

2. Associative property

3. Identity element

4. Inverse element

Example:

(Z, +) → Group

Integers under addition

2) Properties of Group

• Identity unique असतो

• Inverse unique असतो

3) Subgroup

Definition:

Group चा subset जो स्वतः group असतो

Example:

Even integers is subgroup of integers

4) Cyclic Group

Definition:

Single element पासून complete group generate होतो

Example:

(Z, +)

5) Order of Generator

Definition:

Generator पासून group तयार होण्यासाठी लागणारी elements संख्या

Example:

Order of group Z₆ = 6

🔹 Unit II – Cosets and Normal Subgroups

Hours: 07 | Marks: 07

1) Cosets

Definition:

Group चा subset

Types:

• Left coset

• Right coset

2) Lagrange’s Theorem

Statement:

Order of subgroup divides order of group

Formula:

|G| / |H|

3) Normal Subgroup

Definition:

Left coset = Right coset

Symbol:

H ◁ G

4) Quotient Group

Definition:

Group formed by cosets

Symbol:

G / H

🔹 Unit III – Homomorphism and Isomorphism

Hours: 08 | Marks: 08

1) Homomorphism

Definition:

Structure preserving function

Example:

f(x + y) = f(x) + f(y)

2) Homomorphic Image

Output group

3) Kernel of Homomorphism

Definition:

Elements mapped to identity

4) Isomorphism

Definition:

Two groups structurally same

Symbol:

G ≅ H

5) Fundamental Theorem

Statement:

G / Kernel ≅ Image

6) Second Isomorphism Theorem

7) Third Isomorphism Theorem

🔹 Unit IV – Ring Theory

Hours: 07 | Marks: 07

Definition of Ring

Ring म्हणजे set ज्यामध्ये addition आणि multiplication defined असतात

Example:

(Z, +, ×)

Properties of Ring

Commutative Ring

ab = ba

Ring with Unity

Multiplicative identity exists

Zero Divisor

ab = 0

Ring without Zero Divisor

Integral domain

Subring

Subset which is ring

Integral Domain

No zero divisor

Example:

Integers

Field

Every element has inverse

Example:

Real numbers

Subfield

Subset of field

Prime Field

Basic field

Example:

Rational numbers

CO1. Solve first order differential equations using different techniques.

CO2. Solve higher order differential equations.

CO3. To find the orthogonal trajectories of the curve.

CO4. Describe the different methods to solve second order differential equations.

CO5. Define the Wronskian and explain its significance in the context of linear differential

equations.

CO6. Calculate the total number of permutations for a given finite set

CO7. Apply group and Ring axioms to determine whether a set with a binary operation forms a group

and ring.

CO8. Apply the ordinary differential equation for solving various physical, chemical and daily life

problems