Groups - Introductory Abstract Algebra (Undergraduate Advanced)

This course introduces groups—one of the fundamental structures in abstract algebra. We cover the group axioms, explore classic examples like integers under addition and symmetry groups, and show how groups model structure and transformations in math. Clear, focused, and designed for learners new to abstract algebra.

1

Enrolment valid for 12 months
This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.
[University] Introduction to Abstract Algebra
[University] Introduction to Abstract Algebra
Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Master the foundational structures of modern mathematics. This learning track provides a direct path through abstract algebra, from basic sets to groups, rings, and fields. It delivers the analytical framework essential for advanced theoretical work. This programme is for undergraduate students in mathematics, computer science, or theoretical physics. It is also essential for professionals requiring a rigorous grasp of algebraic structures for work in cryptography, algorithm design, or quantum computing. Construct rigorous proofs and analyse the properties of groups, rings, and fields. This programme directly prepares you for postgraduate studies in pure mathematics and for advanced technical roles in cryptography and algorithm theory.

Course Chapters

1. Introduction

Definition of a group (group axioms), basic properties of groups (uniqueness of identity and inverses, inverse of identity, inverse of inverse, cancellation laws), related structures (groupoid, semigroup, monoid, group, abelian group), order of a group.

No lesson yet.

2. Examples

Examples of groups with verification using group axioms.

No lesson yet.

3. Cayley Tables

Cayley tables for groups, examples of group operations presented in table form.

No lesson yet.

4. Abelian Groups

Definition of abelian groups, Cayley tables, examples of commutative groups.

No lesson yet.

5. Order

Order of a group, order of an element, illustrative examples.

No lesson yet.

6. Residue Classes

Definition of residue classes, operations on residue classes, associated group structures.

No lesson yet.

7. Cyclic Groups

Definition of cyclic groups, key properties, examples of cyclic groups.

No lesson yet.

8. Permutation Groups

Definition of permutation groups, composition of permutations, structure of groups formed by permutations.

No lesson yet.