2021 Faculty Courses School of Science Undergraduate major in Mathematics
Introduction to Algebra IV
- Academic unit or major
- Undergraduate major in Mathematics
- Instructor(s)
- Yuri Yatagawa / Shane Kelly / Tatsuhiro Minagawa
- Class Format
- Lecture/Exercise
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-8 Fri (H112)
- Class
- -
- Course Code
- MTH.A204
- Number of credits
- 110
- Course offered
- 2021
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 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 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 excercises are given related to the contents of the preceding lecture. This course succeeds ``Introduction to Algebra III'' offered in the third quater.
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 familiar with important notions such as homomorphisms of groups, normal subgroups, the fundamental theorem on group homomorphisms, conjugacy classes, class equation, and actions of groups.
To become able to prove by him/herself basic properties of these objects.
Keywords
homomorphism of groups, normal subgroup, the fundamental theorem on group homomorphisms, conjugacy class, class equation, action of a group
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
Standard lecture course accompanied by discussion sesssions.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Homomorphisms of groups, image and kernel of a homomorphism of groups | Details will be provided during each class session. |
| Class 2 | Discussion session on homomorphisms of groups and image and kernel of a homomorphism of groups | Details will be provided during each class session. |
| Class 3 | Normal subgroups, residue groups | Details will be provided during each class session. |
| Class 4 | Discussion session on normal subgroups and residue groups | Details will be provided during each class session. |
| Class 5 | The first, second and third fundamental theorems on group homomorphisms | Details will be provided during each class session. |
| Class 6 | Discussion session on the first, second and third fundamental theorems on group homomorphisms | Details will be provided during each class session. |
| Class 7 | Subgroups generated by subsets | Details will be provided during each class session. |
| Class 8 | Discussion session on subgroups generated by subsets | Details will be provided during each class session. |
| Class 9 | Conjugacy of elements, conjugacy classes, centralizers | Details will be provided during each class session. |
| Class 10 | Discussion session on conjugacy of elements, conjugacy classes, and centralizers | Details will be provided during each class session. |
| Class 11 | Class equation and its applications | Details will be provided during each class session. |
| Class 12 | Discussion session on the class equation and its applications | Details will be provided during each class session. |
| Class 13 | Actions of groups | Details will be provided during each class session. |
| Class 14 | Discussion session on actions of groups | Details will be provided during each class session. |
| Class 15 | Checking 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)
Shoichi Nakajima : Basics of Algebra and Arithmetic, Kyoritsu Shuppan, Co., Ltd., 2000.
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 a lecture.
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.