Course title
V01305002
Algebra 1

 TAMORI Hiroyoshi
Course description
We learn basics of group theory.
Purpose of class
We learn fundamental concepts in group theory. This course is prepared as a first course in abstract algebra. We study abstract group theory together with several 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.
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" 200minutes
Class assignment
2. Modular arithmetic Reviewing the previous class 200minutes
Class assignment
3. Extended Euclidean algorithm, Multiplicative group of integers modulo n, Fermat's little theorem Reviewing the previous class 200minutes
Class assignment
4. Symmetric group Reviewing the previous class 200minutes
Class assignment
5. Dihedral group Reviewing the previous class 200minutes
Class assignment
6. Subgroup generated by a subset, Order of an element of groups, Cyclic group Reviewing the previous class 200minutes
Class assignment
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 200minutes
Class assignment
9. Lagrange's theorem and its application, Normal subgroup, Quotient group Reviewing the previous class 200minutes
Class assignment
10. Group homomorphism, Group isomorphism, Image, Kernel Reviewing the previous class 200minutes
Class assignment
11. Fundamental theorem on homomorphisms,
Direct product of groups, Chinese remainder theorem 		Reviewing the previous class 200minutes
Class assignment
12. Group action, G-orbit, Stabilizer subgroup, G-orbit decomposition, Orbit-stabilizer theorem Reviewing the previous class 200minutes
Class assignment
13. Commutator subgroup, Solvable group, Sylow's theorem Reviewing the previous class 200minutes
Class assignment
14. Final exam and comments on it Reviewing the previous classes 200minutes
Total. - - 2800minutes
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Class assignments Mid-term exam Final exam Total.
1. 8% 8% 8% 24%
2. 8% 8% 8% 24%
3. 8% 10% 8% 26%
4. 8% 8% 10% 26%
Total. 32% 34% 34% -
Evaluation method and criteria
Students are evaluated by Class assignments (about 33%), Mid-term exam (about 33%), and Final exam (about 33％). 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"
Feedback in 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
The fundamental notions studied in the class "Fundamentals of Mathematics", such as sets and maps.
Office hours and How to contact professors for questions
  • Office Hours: Wednesday 12:30-13:20
  • You can ask questions before/after each lecture.
  • 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 : Sat Sep 09 08:32:29 JST 2023