2020 Students Enrolled in or before 2015 School of Science Mathematics
Exercises in Algebra A II
- Academic unit or major
- Mathematics
- Instructor(s)
- Satoshi Naito / Yuri Yatagawa / Shane Kelly / Mutsuro Somekawa
- Class Format
- Exercise (Zoom)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-8 Fri (H112)
- Class
- -
- Course Code
- ZUA.A204
- Number of credits
- 020
- Course offered
- 2020
- Offered quarter
- 3-4Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This course is an exercise session for "Introduction to Algebra II'' (ZUA.A203). The materials for exercise are chosen from that course.
Course description and aims
To become familiar with important notions such as the axiom of groups, subgroups, residue classes, order, cyclic groups, symmetric groups, 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
group, subgroup, residue class, order, cyclic group, symmetric group, 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
Students are given exercise problems related to what is taught in the course "Introduction to Algebra II'''.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Discussion session on the axiom of groups, typical examples of groups, first properties of groups | Details will be announced during each lecture. |
| Class 2 | Discussion session on basic properties of the operation in a group and of the identity and inverse elements | Details will be announced during each lecture. |
| Class 3 | Discussion session on the definition of a subgroup, criterion for subgroups, and examples of subgroups | Details will be announced during each lecture. |
| Class 4 | Discussion session on right- and left-cosets by a subgroup | Details will be announced during each lecture. |
| Class 5 | Discussion session on the order of a group and Lagrange's theorem | Details will be announced during each lecture. |
| Class 6 | Discussion session on the order of an element of a group and on cyclic groups | Details will be announced during each lecture. |
| Class 7 | Discussion session on symmetric groups | Details will be announced during each lecture. |
| Class 8 | Discussion session on homomorphisms of groups and image and kernel of a homomorphism of groups | Details will be announced during each lecture. |
| Class 9 | Discussion session on normal subgroups and residue groups | Details will be announced during each lecture. |
| Class 10 | Discussion session on the first, second and third fundamental theorems on group homomorphisms | Details will be announced during each lecture. |
| Class 11 | Discussion session on subgroups generated by subsets | Details will be announced during each lecture. |
| Class 12 | Discussion session on conjugacy of elements, conjugacy classes, and centralizers | Details will be announced during each lecture. |
| Class 13 | Discussion session on the class equation and its applications | Details will be announced during each lecture. |
| Class 14 | Discussion session on actions of groups | Details will be announced during each lecture. |
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
Brief exam and oral presentation for exercise problems. Details will be announced during a lecture.
Related courses
- MTH.A203 ： Introduction to Algebra III
- MTH.A204 ： Introduction to Algebra IV
- ZUA.A201 ： Introduction to Algebra I
- ZUA.A202 ： Exercises in Algebra A I
- ZUA.A203 ： Introduction to Algebra II
Prerequisites
Students are supposed to have completed [Linear Algebra I / Recitation], [Linear Algebra II], [Linear Algebra Recitation II], [Introduction to Algebra I (ZUA.A201)] and [Exercises in Algebra A I (ZUA.A202)].
Students are strongly recommended to take ZUA.A203: Introduction to Algebra II (if not passed yet) at the same time.