Course title
330151002
Algebra 1

 TAMORI Hiroyoshi
Middle-level Diploma Policy (mDP)
Program / Major mDP Goals
IoT Course DP-4a・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Software Course DP-4b・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Media Course DP-4c・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Data Science Course DP-4d・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Mechatronics Course DP-4・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Architecture and Architectural Engineering Course DP-4a・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Environmental Systems and Urban Planning Course DP-4b・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Bioscience Course DP-4a・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Biomedical Engineering Course DP-4b・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Sports Engineering Course DP-4c・3 専門分野と他分野を関連付ける素養
主軸となる分野の専門知識を他分野と関連付ける分野横断型の知識と行動力を修得し、社会で活用できる。
Mathematical Sciences Course DP-4・1 研究者・技術者としての基礎的素養
代数、幾何、解析、応用数学、情報数学などの専門知識を修得し、課題を抽象化して数理的な問題設定をすると共に、適切な理論的考察・計算的手法を用いて問題解決策を構築できる。
Purpose of class
We learn basics of group theory. We learn how to study abstract mathematics together by using concrete examples.
Course description
A set equipped with a binary operation where associativity holds, an identity element exists, and every element has an inverse is called a group. We learn several concrete examples of group, fundamental theorems in group theory, and their applications to concrete examples. This course is prepared as a first course in abstract algebra.
Goals and objectives
  1. To be able to explain the definitions of group, subgroup, homomorphism, coset, quotient group, and group action.
  2. To be able to explain examples of groups, subgroups, homomorphisms, cosets, quotient groups, and group actions.
  3. To be able to perform calculations in integer residue rings, symmetric groups, dihedral groups, and matrix groups.
  4. To understand theorems in group theory, and to be able to apply them to concrete examples.
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Class assignments Mid-term exam Final exam Total.
1. 7% 8% 10% 25%
2. 8% 7% 10% 25%
3. 7% 8% 10% 25%
4. 8% 7% 10% 25%
Total. 30% 30% 40% -
Evaluation method and criteria
Students are evaluated by Class assignments (about 30%), Mid-term exam (about 30%), and Final exam (about 40％). One criterion for earning credits is to be able to give concrete examples of groups, and to be able to explain theorems in group theory by using concrete examples.
Language
Japanese
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1. Guidance, Group, Subgroup Reviewing logic and linear algebra 100minutes
Class assignment 100minutes
2. Dihedral group, Modular arithmetic Reviewing the previous class 100minutes
Class assignment 100minutes
3. Extended Euclidean algorithm, Multiplicative group of integers modulo n, Fermat’s little theorem Reviewing the previous class 100minutes
Class assignment 100minutes
4. Symmetric group Reviewing the previous class 100minutes
Class assignment 100minutes
5. Equivalence relation, Equivalence class, Representative, Quotient set, Left and right coset, Index of a subgroup, Order of an element of groups Reviewing the previous class 100minutes
Class assignment 100minutes
6. Lagrange’s theorem and its application, Subgroup generated by a subset, Cyclic group, Normal subgroup, Quotient group Reviewing the previous class 100minutes
Class assignment 100minutes
7. Mid-term exam Reviewing the previous classes 200minutes
8. Group homomorphism, Group isomorphism, Image, Kernel Reviewing the previous class 100minutes
Class assignment 100minutes
9. Fundamental theorem on homomorphisms Reviewing the previous class 100minutes
Class assignment 100minutes
10. Direct product of groups, Chinese remainder theorem, Group action Reviewing the previous class 100minutes
Class assignment 100minutes
11. G-orbit, Stabilizer subgroup, G-orbit decomposition, Orbit-stabilizer theorem Reviewing the previous class 100minutes
Class assignment 100minutes
12. Supplement and Exercises Reviewing the previous class 100minutes
Class assignment 100minutes
13. Commutator subgroup, Solvable group, Sylow’s theorem Reviewing the previous class 100minutes
Class assignment 100minutes
14. Final exam Reviewing the previous classes 200minutes
Total. - - 2800minutes
Feedback on exams, assignments, etc.
ways of feedback specific contents about "Other"
Feedback in/outside the class.
Textbooks and reference materials
【Reference book】
Akihiko YUKIE ”代数学1　群論入門” (Nippon Hyoron sha co.,Ltd. ISBN-978-4-535-78659-2)
Toshiyuki KATSURA ”代数学I　 群と環” (University of Tokyo Press, ISBN-978-4-13-062951-5)

Students do not need to purchase the above reference books.
Prerequisites
Office hours and How to contact professors for questions
  • You can ask questions before/during/after each lecture.
  • Office Hours: Wednesday 12:30-13:20
  • You can visit the office of the lecturer and ask questions.
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Course that cultivates a basic self-management skills
Active-learning course
About half of the classes are interactive
Course by professor with work experience
Work experience Work experience and relevance to the course content if applicable
N/A N/A
Education related SDGs:the Sustainable Development Goals
  • 4.QUALITY EDUCATION
  • 9.INDUSTRY, INNOVATION AND INFRASTRUCTURE
Last modified : Mon Mar 23 04:07:20 JST 2026