Subgroups - Introductory Abstract Algebra (Undergraduate Advanced)
[University] Introduction to Abstract AlgebraMaster 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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