Pedagogical Mapping

Undergraduate (UG)

A structured visualization of mathematical curriculum units categorized by subject domain for the Undergraduate (UG) level.

140

Total Units

Subject Domains

Topic

Abstract Algebra

Chapter 01

Introduction to Groups

Chapter 02

Subgroups and Normal Subgroups

Subgroups

Unit 01

Academic Node

Finite Subgroups

Explore Module

Unit 02

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Intersections of Subgroups

Explore Module

Unit 03

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Products of Subgroups

Explore Module

Unit 04

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Subgroup Criteria

Explore Module

Unit 05

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Subgroups

Explore Module

Unit 06

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Unions of Subgroups

Explore Module

Special Subgroups

Unit 01

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Centraliser of an Element

Explore Module

Unit 02

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Centre and Centralisers

Explore Module

Unit 03

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Centre of a Group

Explore Module

Cosets of Subgroups

Unit 01

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Cardinality and Correspondence of Cosets

Explore Module

Unit 02

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Coset Membership and Equality Criteria

Explore Module

Unit 03

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Cosets and Intersections of Subgroups

Explore Module

Unit 04

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Cosets as Equivalence Classes

Explore Module

Unit 05

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Cosets as Partitions

Explore Module

Unit 06

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Cosets Determined by Elements of a Subgroup

Explore Module

Unit 07

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Equal and Disjoint Cosets

Explore Module

Unit 08

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Left and Right Cosets

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Index and Lagrange Theorem

Unit 01

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Divisors and Subgroup Orders

Explore Module

Unit 02

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Groups of Prime Order

Explore Module

Unit 03

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Index of a Subgroup

Explore Module

Unit 04

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Lagrange Theorem

Explore Module

Unit 05

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Order of an Element in a Finite Group

Explore Module

Unit 06

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Powers in a Finite Group

Explore Module

Applications of Lagrange Theorem

Unit 01

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Index and Intersections of Subgroups

Explore Module

Unit 02

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Index Multiplication Formula

Explore Module

Unit 03

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Product Formula for Finite Subgroups

Explore Module

Unit 04

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Relatively Prime Subgroup Orders

Explore Module

Unit 05

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Subgroups of Prime Index

Explore Module

Unit 06

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Subgroups of Prime Power Order Groups

Explore Module

Double Cosets

Unit 01

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Cardinality of Double Cosets

Explore Module

Unit 02

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Double Coset Definition

Explore Module

Unit 03

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Double Cosets as Equivalence Classes

Explore Module

Normal Subgroups

Unit 01

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Central and Cyclic Sources of Normal Subgroups

Explore Module

Unit 02

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Definition and First Examples of Normal Subgroups

Explore Module

Unit 03

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Index Two and Coset Criteria for Normality

Explore Module

Unit 04

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Normalisers of Subgroups

Explore Module

Unit 05

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Normality Inside Related Subgroups

Explore Module

Unit 06

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Products and Intersections of Normal Subgroups

Explore Module

Quotient Groups

Unit 01

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Abelian and Cyclic Quotient Groups

Explore Module

Unit 02

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Abelian Quotients and Commutators

Explore Module

Unit 03

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Construction of Quotient Groups

Explore Module

Unit 04

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Normal Subgroups in Quotient Groups

Explore Module

Unit 05

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Order of a Quotient Group

Explore Module

Unit 06

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Quotients by Central Subgroups

Explore Module

Chapter 03

Homomorphisms and Isomorphisms of Groups

Homomorphisms of Groups

Unit 01

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Homomorphism of Groups

Explore Module

Unit 02

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Basic Properties of Homomorphisms

Explore Module

Unit 03

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Kernel of a Homomorphism

Explore Module

Unit 04

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Epimorphism, Monomorphism, Isomorphism, and Automorphism

Explore Module

Unit 05

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Natural Homomorphism

Explore Module

Unit 06

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Isomorphism Tests and Examples

Explore Module

Unit 07

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Classification of Cyclic Groups

Explore Module

Unit 08

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Homomorphisms Between Cyclic Groups

Explore Module

Unit 09

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Power Maps on Groups

Explore Module

Unit 10

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Cayley Theorem and Permutation Groups

Explore Module

Isomorphism and Correspondence Theorems

Unit 01

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Factorization Through Quotients

Explore Module

Unit 02

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First Isomorphism Theorem

Explore Module

Unit 03

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Homomorphic Images and Normal Subgroups

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Unit 04

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Second Isomorphism Theorem

Explore Module

Unit 05

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Third Isomorphism Theorem

Explore Module

Unit 06

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Correspondence Theorem

Explore Module

Group Actions

Unit 01

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Group Actions and G-Sets

Explore Module

Unit 02

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Orbits and Stabilizers

Explore Module

Unit 03

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Orbit-Stabilizer Formula and Orbit Counting

Explore Module

Unit 04

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Permutation Representations and Extended Cayley Theorem

Explore Module

Unit 05

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Fixed Points and Burnside's Lemma

Explore Module

Chapter 04

Sylow Theorems

Conjugacy Classes

Unit 01

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Centralizers and Conjugacy of Elements

Explore Module

Unit 02

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Class Equation and Conjugacy Class Decomposition

Explore Module

Unit 03

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Conjugate Subgroups and Invariance

Explore Module

Unit 04

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Normalizers and Counting Conjugate Subgroups

Explore Module

Cauchy's Theorem and p Groups

Unit 01

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Cauchy's Theorem

Explore Module

Unit 02

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Subgroups of Finite Abelian Groups

Explore Module

Unit 03

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p-Groups and p-Subgroups

Explore Module

Unit 04

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Centers of Finite p-Groups

Explore Module

Unit 05

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Groups of Order p Squared and p Cubed

Explore Module

Unit 06

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Subgroups of Finite p-Groups

Explore Module

Unit 07

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Normal Subgroups from Cauchy's Theorem

Explore Module

Unit 08

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Computing p-Subgroups

Explore Module

Sylow Theorems

Unit 01

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Sylow's First Theorem

Explore Module

Unit 02

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Sylow p-Subgroups

Explore Module

Unit 03

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Conjugates and Normality of Sylow p-Subgroups

Explore Module

Unit 04

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Sylow's Second Theorem

Explore Module

Unit 05

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Sylow's Third Theorem

Explore Module

Unit 06

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Sylow Subgroups in Normal Subgroups and Quotients

Explore Module

Unit 07

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Applications of Sylow Counting

Explore Module

Unit 08

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Direct Products of Sylow Subgroups

Explore Module

Some Applications of the Sylow Theorems

Unit 01

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Simple Groups and Non-Simplicity Tests

Explore Module

Unit 02

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Non-Simplicity of Groups of Small Order

Explore Module

Unit 03

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Simple Groups of Order 60 and A5

Explore Module

Unit 04

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Groups of Order pq

Explore Module

Unit 05

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Dihedral Groups and Groups of Order 2p

Explore Module

Unit 06

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Groups of Small Order

Explore Module

Unit 07

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Cyclicity from Sylow Subgroups

Explore Module

Unit 08

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Central Sylow Subgroups and Normality Applications

Explore Module

Unit 09

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True-or-False Applications of Sylow Theorems

Explore Module

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

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

Centre and Centralisers

Centre of a Group

Cosets of Subgroups

Cardinality and Correspondence of Cosets

Coset Membership and Equality Criteria

Cosets and Intersections of Subgroups

Cosets as Equivalence Classes

Cosets as Partitions

Cosets Determined by Elements of a Subgroup

Equal and Disjoint Cosets

Left and Right Cosets

Index and Lagrange Theorem

Divisors and Subgroup Orders

Groups of Prime Order