### 2021　Exercises in Algebra A II

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Mathematics
Instructor(s)
Yatagawa Yuri  Kelly Shane  Minagawa Tatsuhiro
Course component(s)
Exercise    (ZOOM)
Day/Period(Room No.)
Fri5-8(H112)
Group
-
Course number
ZUA.A204
Credits
2
2021
Offered quarter
3-4Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
Japanese
Access Index

### Course description and aims

This course is an exercise session for "Introduction to Algebra II'' (ZUA.A203). The materials for exercise are chosen from that course.

### Student learning outcomes

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 that will be developed

 ✔ 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/Required learning

Course schedule Required learning
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.

### Out-of-Class Study Time (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.

### Assessment criteria and methods

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 (i.e., required knowledge, skills, courses, etc.)

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.