YongTao Chen
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BTG Basic Theory of Groups

  • Mr. Chen YongTao
Course subject : Math: Algebra
Language : English (US)
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 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. 

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 exercises provide an opportunity to apply the concepts and techniques presented, and the homeworks are designed with the goal of testing the understanding of corresponding topics. The present course should be suitable for both mathematical major undergraduates and the students majored in other areas which involves the applications of group theory.

The course consists of 8 lectures, each of which contains two periods of lessons. In this course you will be introduced to basic concepts and techniques used in group theory. The specific content of the present course is indicated by the titles of the lectures, which are listed as follows:

  1. Concept and basic properties of groups;
  2. Subgroups and Orders;
  3. Cyclic groups;
  4. Cosets and Lagrange's Theorem;
  5. Normal subgroups, Quotient groups;
  6. Group homomorphisms and isomorphisms;
  7. Permutation Groups;
  8. Group Actions.

A set of review problems is provided at the end of the course. 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, the participants may go through the course according to their own schedules.

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 you are 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, you 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.