Foundational Group Theory: Concepts and Applications

Group theory provides the essential framework for understanding symmetry, structure, and abstraction across various mathematical and scientific disciplines. This course delivers a rigorous, text-based introduction to this core branch of abstract algebra. By the end of this course, you will have a deep, working understanding of fundamental group structures, allowing you to solve problems involving permutations, cyclic groups, and basic group properties with confidence. What you'll learn: * Learn the rigorous definitions of groups, subgroups, cosets, and normal subgroups. * Understand foundational theorems, including Lagrange's Theorem and the basics of isomorphism and homomorphism. * Practice manipulating permutation groups and computing with cyclic groups and generators. * Master the structure of quotient groups and direct products of groups. * Apply group theory concepts to introductory problems in public-key cryptography and coding theory. The course begins with core definitions and numerous concrete examples, progresses through essential structural theorems and properties, and concludes with an introduction to applications and advanced concepts like group actions. This course is designed for absolute beginners in abstract algebra, requiring only a basic familiarity with mathematical notation and logical reasoning. Start building your foundational knowledge in pure mathematics today.