Description of Course Goals and CurriculumThe course covers the theory of groups as sets of symmetries. Applications to other fields such as physics, chemistry, computer science, and electrical engineering are frequently discussed as illuminating examples. The goal of the course is to familiarize students with groups and how they can be used to describe more concrete concepts seen in other disciplines. Several core examples of groups are established which appear repeatedly both during the course and in nature. Finite groups are the main focus of the theorems developed in the course. The course can be approximately divided into three parts. The first part, which is the first half of the course until the midterm, focuses on developing a solid foundation of important definitions, followed by exposure to a variety of groups with different properties. There is special focus on cyclic groups and symmetric groups. The second part of the course begins after the definition of group homomorphisms and isomorphisms, which makes available powerful tools for identifying and classifying groups. As a result, the classification of groups is the main focus of this part of the course. For the first two parts, the class essentially follows the textbook, Contemporary Abstract Algebra by Gallian, until the end of the discussion on groups. The final part of the course, approximately corresponding to the last two weeks of instruction, moves from basic group theory to a related topic with applications, in this case representation theory. At this point, the textbook used is Serre’s Linear Representations of Finite Groups. This textbook is a graduate textbook and significantly more difficult to understand than Gallian.
Learning From Classroom InstructionThe course consists of lectures twice a week. There is no review class or precept; instead, Piazza serves as the forum for homework questions and assistance. During class, the professor alternates between giving definitions or stating and proving theorems, and showing examples. Unlike in other math courses, the examples are not usually examples of problem-solving, but literal examples of group properties. The theorem proofs discussed in class and the examples in the textbook are much better representations of the types of problems seen on homework. The examples of groups are instead useful for gaining intuition about group properties. It is important to understand theorems intuitively. The lectures sometimes overlap with the textbook significantly, and sometimes not. The only way to know is to read the relevant sections beforehand. Often, if possible, the professor will try to do proofs in alternate ways from the book, or in a different order. Taking notes in class can be helpful in these situations. This is especially true of the section on representation theory, as the textbook by Serre is extremely terse.
Learning For and From AssignmentsMuch of the learning in the course is achieved through doing problems, as with any kind of math. In addition to serving as practice to increase familiarity with the concepts, the problem sets also develop some theorems and lemmas which are not otherwise taught in the course. There are weekly problem sets, a midterm, and a final. The weekly problem sets consist of problems from the textbook as well as possible supplementary problems set by the professor. The professor gives more or fewer supplementary problems depending on the quality of the problems that can be found in the textbook, and their relevance to the topic at hand. In general, the supplementary problems are somewhat more difficult than the problems in the book. The homework also includes reading from the book. Doing the reading before doing the problem set makes the problems fairly straightforward. However, the problem sets, and the material in general, become more difficult toward the end of the semester. The exams are an accurate representation of the material covered in the course, and on the problem sets in particular. The professor gives exam problems similar to homework problems in both format and difficulty. There are no practice exams, and it suffices to go through the homework, in terms of problem-solving practice.
External ResourcesPiazza is used for question and answer. The professor encourages students to answer each other’s questions, but the professor response time is also good and usually within a day. For more involved discussions, office hours are more appropriate.
What Students Should Know About This Course For Purposes Of Course SelectionThe only official prerequisite to this course is some form of linear algebra. The professor assumes fairly intimate knowledge of basic linear algebra concepts and theorems. As the course is entirely proof-based from the outset, some experience with writing proofs is helpful, and therefore MAT204 or MAT217 are recommended. This course is well-suited to people who are interested in the group theory and its applications. In particular, concepts learned in this course appear frequently in other courses, though not usually by name. Some courses in which concepts from this course are used include PHY208, PHY305, ELE396, and CHM407. Group theoretic concepts are useful in any subject that deals with symmetries.