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.

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.
MTH 203: Sets, Logic and Algebra I
MTH 203: Sets, Logic and Algebra I
Abstract algebra is the structural foundation of modern advanced mathematics. This track delivers the complete NUC CCMAS MTH 203 curriculum, rigorously transitioning you from computational arithmetic to abstract mathematical reasoning. It provides the necessary prerequisite framework for understanding complex mathematical systems and their applications. This programme is targeted at undergraduate students in mathematics, computer science, and physics requiring a firm grounding in fundamental algebraic structures. It also serves professionals in fields like cryptography or theoretical computer science who need a rigorous theoretical refresher. You will master the precise definitions, properties, and relations of core algebraic structures, including groups, subgroups, rings, and fields. You will gain competence in constructing formal proofs using set theory and logic, and understand how homomorphisms preserve mathematical structure. Completion establishes the critical theoretical base demanded for advanced studies in algebra, coding theory, and algorithm design.

Abstract algebra is the structural foundation of modern advanced mathematics. This track delivers the complete NUC CCMAS MTH 203 curriculum, rigorously transitioning you from computational arithmetic to abstract mathematical reasoning. It provides the necessary prerequisite framework for understanding complex mathematical systems and their applications. This programme is targeted at undergraduate students in mathematics, computer science, and physics requiring a firm grounding in fundamental algebraic structures. It also serves professionals in fields like cryptography or theoretical computer science who need a rigorous theoretical refresher. You will master the precise definitions, properties, and relations of core algebraic structures, including groups, subgroups, rings, and fields. You will gain competence in constructing formal proofs using set theory and logic, and understand how homomorphisms preserve mathematical structure. Completion establishes the critical theoretical base demanded for advanced studies in algebra, coding theory, and algorithm design.

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Course Chapters

1. Introduction
Definition of homomorphism and examples.
2. Kernel and Image
Definition, computation, and examples.
3. Properties of Homomorphisms
Preservation of identity and inverses, kernel-image relationships.
4. Isomorphisms
Definition and examples of group isomorphism.
5. Isomorphic Groups
Isomorphism as an equivalence relation, consequences.