Introduction to Group Theory: Foundations of Abstract Algebra

Group theory is the essential language for understanding symmetry and structure across mathematics and science. This course provides a rigorous, step-by-step introduction to this foundational subject. By the end of this course, you will be able to define, analyze, and classify fundamental algebraic structures, preparing you for advanced studies in abstract algebra, topology, and discrete mathematics. What you'll learn: * Understand the axioms defining a group and identify various examples of groups, including cyclic, dihedral, and permutation groups. * Master the concepts of subgroups, cosets, and the powerful applications of Lagrange's Theorem. * Analyze group structure using normal subgroups, quotient groups, and the fundamental Isomorphism Theorems. * Apply the foundational principles of group homomorphisms and isomorphisms to relate different algebraic structures. * Practice calculating with group actions and permutations, essential for applied and computational algebra. The course begins with fundamental definitions and examples, progressing systematically through structural analysis, including theorems on subgroups and quotient groups. The final sections focus on powerful tools like group actions and fundamental mapping principles. This course is designed for absolute beginners in abstract algebra, including students, programmers, and enthusiasts with an interest in mathematical foundations. No prior knowledge of group theory is required. Start your journey into the elegance of abstract algebra today.