Groups - Introductory Abstract Algebra (Undergraduate Advanced)
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[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 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.
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2. Examples
Examples of groups with verification using group axioms.
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3. Cayley Tables
Cayley tables for groups, examples of group operations presented in table form.
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4. Abelian Groups
Definition of abelian groups, Cayley tables, examples of commutative groups.
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5. Order
Order of a group, order of an element, illustrative examples.
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6. Residue Classes
Definition of residue classes, operations on residue classes, associated group structures.
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7. Cyclic Groups
Definition of cyclic groups, key properties, examples of cyclic groups.
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8. Permutation Groups
Definition of permutation groups, composition of permutations, structure of groups formed by permutations.
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