Fundamental theorems from Group theory

Modern algebra is importance for advanced math. It includes group theory, ring theory, and field theory. For the first course of Modern algebra, group theory, is not easy to think. Furthermore, group theorem is also a study of algebraic structure. We will focus set theorem and modular arithmetic a little bit. We also need to know something about functions (injection, surjection, and bijection). We also involve some basic method of proving problem, such as induction. Meanwhile, we summarize fundamental theory for finite groups and some infinite group cases. First, we introduce some basic definitions. Next, we summary some basic types of subgroups. Finally, we prove an Orbit- Stabilizer Theorem.