Group Theory for JAM and Graduate Mathematics Entrance Exams

Group theory is a cornerstone of abstract algebra, yet its highly conceptual nature makes it one of the most challenging topics on graduate mathematics entrance exams. This text-based course guides you step-by-step from foundational definitions to advanced algebraic structures, building the rigorous mathematical intuition needed to solve complex exam problems. What you'll learn: - Understand the fundamental definitions of groups, subgroups, and abelian structures. - Analyze cyclic groups, generators, and the properties of permutation groups. - Apply Lagrange's theorem and explore the significance of cosets and normal subgroups. - Master group homomorphisms, isomorphisms, and the fundamental isomorphism theorems. - Practice solving typical exam-style problems using systematic algebraic proofs and logical deduction. You will begin by mastering essential terminology and basic set-theoretic foundations before progressing systematically through subgroups, cyclic structures, and quotient groups. Each module reinforces theoretical concepts with detailed, written step-by-step proofs and analytical exercises. This course is designed for undergraduate mathematics students preparing for JAM and other graduate-level entrance exams, requiring only a basic familiarity with set theory and matrix algebra. Start mastering abstract algebra today to elevate your exam readiness.