6-Day Intensive Course

ETS 1.0 Abstract Algebra

Learn Abstract Algebra concepts guided by JAM/GATE/JRF PhD experts, having exam-ready syllabus and live sessions.

January 19-24, 2026
Enroll Now
Certificate on Completion
6
Days
12
Sessions
15
Hours Total
100+
Practice Problems

Why Study Algebra?

Algebra is the foundation of modern mathematics, cryptography, and abstract structure theory. This course provides comprehensive coverage of groups, rings, and fields with rigorous proofs, deep theoretical insights, and competition-oriented problem-solving for IIT JAM and CUET PG aspirants.

Group Theory

Cyclic, Abelian, non-Abelian, permutation groups, normal subgroups, quotient groups, and homomorphisms.

Ring Theory

Rings, subrings, ideals, prime ideals, maximal ideals, and quotient ring construction.

Field Theory

Introduction to fields and quotient fields.

Your Learning Journey

From basic group structures to quotient fields and advanced algebra

Group Theory Foundations

Cyclic, abelian, non-abelian, and permutation groups with their fundamental properties.

Normal Subgroups & Quotient Groups

Master Lagrange's theorem, quotient structure, and group homomorphisms.

Ring Theory Essentials

Rings, subrings, and ideals—prime ideals and maximal ideals with applications.

Field Theory & Quotient Fields

Introduction to fields and quotient field.

Competition & Proof Mastery

Rigorous proof writing and rapid problem-solving for IIT JAM & CUET PG exams.

Certificate of Completion

Earn a verified certificate showcasing your mastery of Algebra

Recognized by academic institutions & research organizations

Comprehensive Curriculum

Six intensive modules covering groups, rings, and fields

Module 1: Foundations of Group Theory

1 Groups & Basic Properties

  • • Definition and examples
  • • Closure, associativity, identity, inverses
  • • Order of elements

2 Subgroups

  • • Definition and subgroup criteria
  • • Generated subgroups
  • • Index of subgroup

3 Abelian vs Non-Abelian

  • • Commutative operations
  • • Examples of non-abelian groups
  • • Center of a group
Competition Focus

Rapid group identification and order computation

Module 2: Subgroups & Permutation Groups

1 Cyclic Groups

  • • Order of cyclic groups
  • • Zn and properties

2 Permutation Groups

  • • Symmetric groups Sn and Alternating groups An
  • • Cycle notation and composition
Competition Focus

Permutation group computations and cycle notation

Module 3: Normal Subgroups & Lagrange's Theorem

1 Cosets

  • • Left and right cosets
  • • Coset decomposition
  • • Index equals number of cosets

2 Lagrange's Theorem

  • • Statement: |H| divides |G|
  • • Order of elements divides group order
  • • Converse failure and Cauchy's theorem

3 Normal Subgroups

  • • Definition: gHg-1 = H
  • • Characterization and criteria
  • • Kernel of homomorphisms
Competition Focus

Lagrange's theorem applications and normal subgroup identification

Module 4: Quotient Groups & Homomorphisms

1 Quotient Groups

  • • G/N construction
  • • Quotient group structure
  • • Order of quotient group

2 Group Homomorphisms

  • • \phi(ab) = \phi(a)\phi(b)
  • • Kernel and image
  • • Homomorphism properties

3 Isomorphism Theorems

  • • First isomorphism theorem
  • • Second and third isomorphism theorems
  • • Correspondence theorem
Competition Focus

Homomorphism identification and isomorphism theorem applications

Module 5: Introduction to Ring Theory

1 Rings & Subrings

  • • Ring definition and properties
  • • Subrings and subring criteria
  • • Examples:Z, Q, R, Zn

2 Ring Homomorphisms

  • • Definition and properties
  • • Kernel of ring homomorphism
  • • Composition of homomorphisms

3 Units & Zero Divisors

  • • Invertible elements (units)
  • • Zero divisors
  • • Integral domains and division rings
Competition Focus

Ring structure identification and unit/zero divisor computation

Module 6: Ideals, Fields & Quotient Structures

1 Ideals

  • • Left, right, two-sided ideals
  • • Principal ideals
  • • Ideal operations and generation

2 Prime & Maximal Ideals

  • • Prime ideal definition
  • • Maximal ideal characterization
  • • Relationship to quotient rings

3 Fields & Quotient Fields

  • • Field definition and properties
  • • Quotient field construction
  • • Integral domains and field of fractions
  • • Characteristic of fields
Competition Focus

Prime/maximal ideal identification and quotient field construction

Course Features

Competition-Oriented

IIT JAM & CUET PG focused with previous year problem analysis

Rigorous Proofs

Complete proofs and deep theoretical foundations

Algebraic Structures

Groups, rings, fields—comprehensive structure coverage

Quotient Structures

Quotient groups, rings, and field theory

Live Interactive Sessions

Real-time proof walkthroughs and problem-solving

100+ Practice Problems

Comprehensive problem sets with detailed solutions

Learn from the Best

Your instructors are IIT JAM, GATE, and CSIR JRF qualified PhD holders with extensive academic and research backgrounds from India's premier institutions.

Target Audience

IIT JAM Mathematics aspirants
CUET PG Mathematics candidates
MSc Mathematics/Physics students
Students pursuing pure mathematics research

Prerequisites

Set theory and basic logic
Mathematical proof-writing experience
Mathematical maturity and abstraction

Perfect for students comfortable with abstract mathematical thinking

Career Pathways

Where algebra expertise leads

Pure Mathematics

Academic research in algebra

Cryptography

Information security specialist

Coding Theory

Error correction & data science

PhD Programs

Advanced research pathways

Enroll Today

Master Algebra for competitive exams and research excellence

₹950
All inclusive • Certificate included
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Problem Sets
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