Course title
V01305002
Algebra 1

 TAMORI Hiroyoshi
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.
Purpose of class
We learn basics of group theory. We learn how to study abstract mathematics together by using concrete examples.
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% -
Language
Japanese
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1. Guidance, Group, Subgroup Reviewing the class "Fundamentals of Mathematics" 100minutes
Class assignment 100minutes
2. 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. Dihedral group Reviewing the previous class 100minutes
Class assignment 100minutes
6. Subgroup generated by a subset, Order of an element of groups, Cyclic group Reviewing the previous class 100minutes
Class assignment 100minutes
7. Mid-term exam and comments on it Reviewing the previous classes 200minutes
8. Equivalence relation, Equivalence class, Representative, Quotient set, Left and right coset, Index of a subgroup Reviewing the previous class 100minutes
Class assignment 100minutes
9. Lagrange's theorem and its application, Normal subgroup, Quotient group Reviewing the previous class 100minutes
Class assignment 100minutes
10. Group homomorphism, Group isomorphism, Image, Kernel Reviewing the previous class 100minutes
Class assignment 100minutes
11. Fundamental theorem on homomorphisms,
Direct product of groups, Chinese remainder theorem 		Reviewing the previous class 100minutes
Class assignment 100minutes
12. Group action, G-orbit, Stabilizer subgroup, G-orbit decomposition, Orbit-stabilizer theorem 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 and comments on it Reviewing the previous classes 200minutes
Total. - - 2800minutes
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.
Feedback on exams, assignments, etc.
ways of feedback specific contents about "Other"
授業内と授業外でフィードバックを行います。
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
The fundamental notions studied in the class "Fundamentals of Mathematics", such as sets and maps.
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 an ability for utilizing knowledge
  • Course that cultivates a basic problem-solving skills
Active-learning course
More than one class is 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 : Wed Mar 19 04:10:17 JST 2025