2025 (Current Year) Faculty Courses School of Science Undergraduate major in Mathematics
Introduction to Algebra IV
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Kazuma Shimomoto / Tatsuhiro Minagawa
- Class Format
- Lecture/Exercise
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.A204
- Number of credits
- 110
- Course offered
- 2025
- Offered quarter
- 4Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
Algebra is a discipline of mathematics that deals with abstract notions which generalize algebraic operations on various mathematical objects. The main subjects of this course include basic notions and properties of groups, which are a mathematical object having just one operation. To help deeper understanding of the newly learnt concepts, each even-numbered class is devoted to a discussion session, where exercises are given related to the contents of the preceding lecture. This course succeeds ``Introduction to Algebra III'' offered in the third quarter.
The theory of groups is a basic language in mathematics and related sciences, and has an extremely wide variety of applications. To exploit groups effectively, however, one needs to be familiar with many concrete examples of groups, not just having a grasp of them as an abstract notion. In this course, typical examples of groups will be provided as well as an abstract treatment of groups based on the notions of sets and maps.
Course description and aims
To become able to explain important notions such as actions of groups, orbits, conjugacy classes, class equations, Sylow theorems, solvable groups, representations of finite groups, and character of representations, together with their examples.
To become able to prove basic properties of these objects by him/herself.
Keywords
actions of groups, orbits, conjugacy classes, class equations, Sylow theorems, solvable groups, representations of finite groups, character of representations
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course accompanied by discussion sessions.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Definition of actions of groups and their examples, stabilizers, orbits, orbit decompositions | Details will be provided during each class session. |
| Class 2 | Discussion session | Details will be provided during each class session. |
| Class 3 | Conjugate by an element of a group, conjugacy classes, class equations | Details will be provided during each class session. |
| Class 4 | Discussion session | Details will be provided during each class session. |
| Class 5 | Application of actions of groups, Sylow theorems | Details will be provided during each class session. |
| Class 6 | Discussion session | Details will be provided during each class session. |
| Class 7 | Solvable groups | Details will be provided during each class session. |
| Class 8 | Discussion session | Details will be provided during each class session. |
| Class 9 | Definition of representations of finite groups and their examples | Details will be provided during each class session. |
| Class 10 | Discussion session | Details will be provided during each class session. |
| Class 11 | Schur's lemma, Maschke's theorem | Details will be provided during each class session. |
| Class 12 | Discussion session | Details will be provided during each class session. |
| Class 13 | Character of representations | Details will be provided during each class session. |
| Class 14 | Discussion session | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None.
Reference books, course materials, etc.
P.J. Cameron : Introduction to Algebra (second ed.), Oxford Univ. Press, 2008.
N. Jacobson : Basic Algebra I (second ed.), Dover，1985.
M. Artin : Algebra (second ed.), Addison-Wesley, 2011.
N. Herstein: Topics in algebra, John Wiley & Sons, 1975.
A. Weil: Number Theory for Beginners, Springer-Verlag, 1979.
Evaluation methods and criteria
Based on evaluation of the results for discussion session and final examination. Details will be announced during lectures.
Related courses
- MTH.A201 ： Introduction to Algebra I
- MTH.A202 ： Introduction to Algebra II
- MTH.A203 ： Introduction to Algebra III
Prerequisites
Students are supposed to have completed [Linear Algebra I / Recitation], [Linear Algebra II], [Linear Algebra Recitation II], [Introduction to Algebra I], [Introduction to Algebra II] and [Introduction to Algebra III].
Other
None in particular.