YongTao Chen has a MSc in pure mathematics, and has been an university math teacher for over two decades. His teaching focuses on Advanced Calculus, Linear Algebra, Abstract Algebra, Real Analysis, and Elementary Number Theory. He also taught Mathematical Logic, Introductory Probability, and Category Theory. Since the year 2011, he has been teaching online mathematical courses with enthusiasm.

The theory of groups can be considered the study of symmetry. Symmetry is not a number or a shape, but a special kind of transformation - a way to move an object. Young French mathematician Galois invented a language known as "group theory" to describe symmetry in mathematical structures, and to deduce its consequences. Today Group Theory is an important branch of modern algebra, with various applications to other disciplines, both inside and outside mathematics, such as geometry, topology, number theory, cryptography, chemistry and physics.

The present course offers a concise introduction to group theory for undergraduate students, and all others with an interest in the subject. The course includes the standard topics taught in typical undergraduate courses on group theory. Each lecture presents the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected examples and exercises. The subject is treated in relatively abstract way so as to make the text as compact as possible, which, I believe, makes the learning more effective at least psychologically. The lecture notes focuses on the importance of rigorous proofs of mathematical statements so as to foster the readers' mathematical maturity. But care has been taken to lead the reader through the proofs by gentle stages, in the hope of making readers comfortable with mathematical thinking and rigorous arguments.

Basic knowledge about elementary number theory and classical set theory is expected, which means that some concepts involved in these two fields will appear in this course. If some participant is feeling rusty or uncertain, I'd encourage you to sign up for those two online courses to take a look. You might remember more than you think! In addition, participants are expected to work at least 6 hours each week. An advice: the experience indicates that reading mathematics is not like reading novels or history. You need to think slowly about every sentence presented in the lecture notes. Usually, you will need to reread the same material later, often more than one rereading.

The course consists of 8 lectures, each of which contains two periods of lessons. The specific content of present course is indicated by the titles of the lectures. Every lecture is followed by a problem set used for home work. A set of review problems is provided at the end of the course so as to test the participants' understanding of the whole course. The exercises provide an opportunity to apply the concepts and techniques just learned, and the home works are designed with the goal of testing the participants' understanding of corresponding topics.

In the spirit of teachings in reality, in each week are delivered two lectures, and the lecturing on the entire course is thus completed in 4 weeks. Certainly, participants may go through the course at their own paces.

#### 1st Week-Lectutre 1: Concept and basic properties of groups

#### 1st Week-Lecture 2: Subgroups and Orders

#### 2nd Week-Lecture 3: Cyclic groups

#### 2nd Week-Lecture 4: Cosets and Lagrange's Theorem

#### 3rd Week-Lecture 5: Normal subgroups, Quotient groups

#### 3rd Week-Lecture 6: Group homomorphisms and isomorphisms

#### 4th Week-Lecture 7: Permutation Groups

#### 4th Week-Lecture 8: Group actions

#### Review Problems