Homomorphisms - Introductory Abstract Algebra (Undergraduate Advanced)

This course explores homomorphisms—the maps that connect algebraic structures while preserving their operations. You’ll learn what makes a function a homomorphism, see examples between groups and rings, and understand how these mappings reveal similarities and relationships between structures. Clear, focused, and designed for first-time learners ready to deepen their understanding of algebraic connections.

Payment required for enrolment
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 homomorphism and examples.

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2. Kernel and Image

Definition, computation, and examples.

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3. Properties of Homomorphisms

Preservation of identity and inverses, kernel-image relationships.

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4. Isomorphisms

Definition and examples of group isomorphism.

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5. Isomorphic Groups

Isomorphism as an equivalence relation, consequences.

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