2024 Students Enrolled in or before 2015 School of Science Mathematics
Exercises in Algebra A II
- Academic unit or major
- Mathematics
- Instructor(s)
- Kazuma Shimomoto / Mutsuro Somekawa
- Class Format
- Exercise (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-8 Fri
- Class
- -
- Course Code
- ZUA.A204
- Number of credits
- 020
- Course offered
- 2024
- Offered quarter
- 3-4Q
- Syllabus updated
- Mar 17, 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 able to explain important notions such as groups, subgroups, orders, cyclic groups, symmetric groups, residue classes, normal subgroups, homomorphisms of groups, the fundamental theorem on group homomorphisms, actions of groups, orbits, conjugacy classes, class equations, Sylow theorems, solvable groups, representations of finite groups, and character of representations, together with their examples.
To become able to prove basic properties of these objects by him/herself.
Keywords
groups, subgroups, orders, cyclic groups, symmetric groups, residue classes, normal subgroups, homomorphisms of groups, the fundamental theorem on group homomorphisms, actions of groups, orbits, conjugacy classes, class equations, Sylow theorems, solvable groups, representations of finite groups, character of representations
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 definition of a group and examples | Details will be announced during each lecture. |
| Class 2 | Discussion session on subgroups | Details will be announced during each lecture. |
| Class 3 | Discussion session on the order of an element of a groups and cyclic groups | Details will be announced during each lecture. |
| Class 4 | Discussion session on symmetric groups | Details will be announced during each lecture. |
| Class 5 | Discussion session on the right- and left-cosets by a subgroup | Details will be announced during each lecture. |
| Class 6 | Discussion session on normal subgroups and quotient groups | Details will be announced during each lecture. |
| Class 7 | Discussion session on homomorphisms of groups and the fundamental theorems on group homomorphisms | Details will be announced during each lecture. |
| Class 8 | Discussion session on the definition of actions of groups and their examples, stabilizers, orbits, and orbit decompositions | Details will be announced during each lecture. |
| Class 9 | Discussion session on conjugate by an element of a group, conjugacy classes, and class equations | Details will be announced during each lecture. |
| Class 10 | Discussion session on application of actions of groups and Sylow theorems | Details will be announced during each lecture. |
| Class 11 | Discussion session on solvable groups | Details will be announced during each lecture. |
| Class 12 | Discussion session on the definition of representations of finite groups and their examples | Details will be announced during each lecture. |
| Class 13 | Discussion session on Schur's lemma and Maschke's theorem | Details will be announced during each lecture. |
| Class 14 | Discussion session on the character of representations | 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)
None.
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 the problem solving situation in the recitation sessions. Details will be provided in the class.
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.
Other
None in particular.