Subgroups - Introductory Abstract Algebra (Undergraduate Advanced)

This course dives into subgroups—key pieces within groups that reveal deeper structure. You’ll learn how to identify subgroups, understand their properties, and see why they matter in the bigger picture of group theory. We’ll also explore examples and criteria that help recognize subgroups quickly. Clear, practical, and perfect for first-time learners ready to explore group structures more deeply.

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 subgroups, notations, and examples.

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2. Subgroup Test

One-step and two-step subgroup criteria, applications.

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3. Trivial and Proper Subgroups

Definition and distinction between trivial, proper, and improper subgroups.

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4. Cyclic Subgroups

Definition, generation, and examples.

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5. Cosets

Definition of left and right cosets, properties and examples.

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6. Lagrange’s Theorem

Statement, proof, and consequences of Lagrange’s theorem.

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