Vidyasagar University · Semester IV

MATHMJ06: Group Theory – I

Pedagogical roadmap structured by subject domains, strictly aligned with the institutional syllabus for MATHMJ06: Group Theory – I.

105

Assigned Units

1

Core Domains

Domain

Abstract Algebra

Chapter 01

Introduction to Groups

Definition and Examples of Groups

Unit 01

Research Module

Additive Groups Number Systems

Analyze Content

Unit 02

Research Module

Chapter 02

Subgroups and Normal Subgroups

Subgroups

Unit 01

Research Module

Finite Subgroups

Analyze Content

Unit 02

Research Module

Intersections of Subgroups

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Unit 03

Research Module

Products of Subgroups

Analyze Content

Unit 04

Research Module

Subgroup Criteria

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Unit 05

Research Module

Subgroups

Analyze Content

Unit 06

Research Module

Unions of Subgroups

Analyze Content

Special Subgroups

Unit 01

Research Module

Centraliser of an Element

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Cosets of Subgroups

Unit 01

Research Module

Cardinality and Correspondence of Cosets

Analyze Content

Index and Lagrange Theorem

Unit 01

Research Module

Divisors and Subgroup Orders

Analyze Content

Applications of Lagrange Theorem

Unit 01

Research Module

Index and Intersections of Subgroups

Analyze Content

Double Cosets

Unit 01

Research Module

Cardinality of Double Cosets

Analyze Content

Normal Subgroups

Unit 01

Research Module

Central and Cyclic Sources of Normal Subgroups

Analyze Content

Quotient Groups

Unit 01

Research Module

Abelian and Cyclic Quotient Groups

Analyze Content

Chapter 03

Homomorphisms and Isomorphisms of Groups

Homomorphisms of Groups

Unit 01

Research Module

Homomorphism of Groups

Analyze Content

Unit 02

Research Module

Basic Properties of Homomorphisms

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Unit 03

Research Module

Kernel of a Homomorphism

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Unit 04

Research Module

Epimorphism, Monomorphism, Isomorphism, and Automorphism

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Unit 05

Research Module

Natural Homomorphism

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Unit 06

Research Module

Isomorphism Tests and Examples

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Unit 07

Research Module

Classification of Cyclic Groups

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Unit 08

Research Module

Homomorphisms Between Cyclic Groups

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Unit 09

Research Module

Power Maps on Groups

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Unit 10

Research Module

Cayley Theorem and Permutation Groups

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Isomorphism and Correspondence Theorems

Unit 01

Research Module

Factorization Through Quotients

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Curated Tracks

Coordinates and Locus

Comprehensive module covering 6 sections in Coordinate Geometry.

Explore Pathway

Invariants under Orthogonal Transformation

Comprehensive module covering 2 sections in Coordinate Geometry.

Explore Pathway

Unit 02

Research Module

Centre and Centralisers

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Unit 03

Research Module

Centre of a Group

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Unit 02

Research Module

Coset Membership and Equality Criteria

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Unit 03

Research Module

Cosets and Intersections of Subgroups

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Unit 04

Research Module

Cosets as Equivalence Classes

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Unit 05

Research Module

Cosets as Partitions

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Unit 06

Research Module

Cosets Determined by Elements of a Subgroup

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Unit 07

Research Module

Equal and Disjoint Cosets

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Unit 08

Research Module

Left and Right Cosets

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Unit 02

Research Module

Groups of Prime Order

Analyze Content

Unit 03

Research Module

Index of a Subgroup

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Unit 04

Research Module

Lagrange Theorem

Analyze Content

Unit 05

Research Module

Order of an Element in a Finite Group

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Unit 06

Research Module

Powers in a Finite Group

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Unit 02

Research Module

Index Multiplication Formula

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Unit 03

Research Module

Product Formula for Finite Subgroups

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Unit 04

Research Module

Relatively Prime Subgroup Orders

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Unit 05

Research Module

Subgroups of Prime Index

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Unit 06

Research Module

Subgroups of Prime Power Order Groups

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Unit 02

Research Module

Double Coset Definition

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Unit 03

Research Module

Double Cosets as Equivalence Classes

Analyze Content

Unit 02

Research Module

Definition and First Examples of Normal Subgroups

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Unit 03

Research Module

Index Two and Coset Criteria for Normality

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Unit 04

Research Module

Normalisers of Subgroups

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Unit 05

Research Module

Normality Inside Related Subgroups

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Unit 06

Research Module

Products and Intersections of Normal Subgroups

Analyze Content

Unit 02

Research Module

Abelian Quotients and Commutators

Analyze Content

Unit 03

Research Module

Construction of Quotient Groups

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Unit 04

Research Module

Normal Subgroups in Quotient Groups

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Unit 05

Research Module

Order of a Quotient Group

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Unit 06

Research Module

Quotients by Central Subgroups

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Unit 02

Research Module

Homomorphic Images and Normal Subgroups

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Unit 03

Research Module

Second Isomorphism Theorem

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Unit 04

Research Module

Third Isomorphism Theorem

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Unit 05

Research Module

Correspondence Theorem

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Kernel of a Homomorphism

27 Jun 2026UG

Abstract Algebra

Introduction to Groups

Definition and Examples of Groups

Additive Groups Number Systems

Binary Operations on Sets

First Nonexamples of Groups

Four Group Axioms

Group Definition

Multiplicative Groups Nonzero Numbers

Verify Group Axioms

Standard Examples of Groups

Additive Groups Modulo N

Alternating Groups

Groups From Number Systems

Klein Four Group

Permutation Groups

Roots of Unity Groups

Semigroup Conditions for Groups

Finite Cancellative Semigroups

Idempotents in Finite Semigroups

Semigroup Equation Criterion for Groups

Semigroups

Unique Idempotent Semigroups

Elementary Properties of Groups

Cancellation Laws in Groups

Equations in Groups

Idempotent Elements in Groups

Identity Element in Groups

Inverse Element in Groups

Properties of Inverses

Abelian Groups

Abelian Group

Elements of Order Two

Groups of Exponent Two

Inverse Product Criterion

Power Criteria for Abelian Groups

Cyclic Groups

Cyclic Group

Cyclic Groups Are Abelian

Even Cyclic Groups

Finite and Infinite Cyclic Groups

Generators of Finite Cyclic Groups

Prime Order Cyclic Groups

Order of an Element

Finite and Infinite Order

Order of Inverse Element

Order of Powers

Orders of Products and Conjugates

Powers Equal to Identity

Prime Order Elements

Powers of Elements in Groups

Conjugation and Powers

Integral Power Laws

Laws of Powers in Groups

Positive and Negative Powers

Power Equations in Groups

Subgroups and Normal Subgroups

Subgroups

Finite Subgroups

Intersections of Subgroups

Products of Subgroups

Subgroup Criteria

Subgroups

Unions of Subgroups

Special Subgroups

Centraliser of an Element

Cosets of Subgroups

Cardinality and Correspondence of Cosets

Index and Lagrange Theorem

Divisors and Subgroup Orders

Applications of Lagrange Theorem

Index and Intersections of Subgroups

Double Cosets

Cardinality of Double Cosets

Normal Subgroups

Central and Cyclic Sources of Normal Subgroups

Quotient Groups

Abelian and Cyclic Quotient Groups

Homomorphisms and Isomorphisms of Groups

Homomorphisms of Groups