1. Group theory studies algebraic structures called groups, which consist of a set $G$ and an operation $\cdot$ combining any two elements $a,b\in G$ to form another element $a\cdot b\in G$. 2. A group must satisfy four axioms: closure ($a\cdot b\in G$), associativity ($(a\cdot b)\cdot c = a\cdot (b\cdot c)$), the existence of an identity element $e$ such that $e\cdot a = a \cdot e = a$, and the existence of inverse elements $a^{-1}$ such that $a\cdot a^{-1} = a^{-1}\cdot a = e$. 3. At its core, group theory abstracts the concept of symmetry and structure, allowing systematic analysis of permutation groups, matrix groups, and more. 4. Real-world applications include cryptography (RSA relies on group properties), physics (symmetry groups describe particle physics), chemistry (molecular symmetry), and computer science (automata theory, coding). 5. Analogies: think of a clock with 12 hours as integers modulo 12 under addition, forming a finite group illustrating cyclic behavior. 6. Advantages: rigorous framework for symmetry, unifying disparate mathematical areas, and powerful tool for solving algebraic equations. 7. Disadvantages: abstractness can be challenging for beginners, some groups are complex to classify fully. This overview builds foundation to explore specific groups and their properties in detail.