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2022　Introduction to Algebra II

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Academic unit or major
Mathematics
Instructor(s)
Yatagawa Yuri 
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Fri3-4(H112)  
Group
-
Course number
ZUA.A203
Credits
2
Academic year
2022
Offered quarter
3-4Q
Syllabus updated
2022/4/20
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

Algebra is a discipline of mathematics that deals with abstract notions which generalize algebraic operations on various mathematical objects. The main subjects of this course include basic notions and properties of groups, which are a mathematical object having just one operation.

The theory of groups is a basic language in mathematics and related sciences, and has an extremely wide variety of applications. To exploit groups effectively, however, one needs to be familiar with many concrete examples, not just having a grasp of them as an abstract notion. In this course, typical examples of groups will be provided as well as an abstract treatment of groups based on the notions of sets and maps.

Student learning outcomes

To become familiar with important notions such as the axiom of groups, subgroups, residue classes, orders, cyclic groups, symmetric groups, homomorphisms of groups, normal subgroups, the fundamental theorem on group homomorphisms, conjugacy classes, class equations, actions of groups, solvable groups, and representations of finite groups.

To become able to prove by him/herself basic properties of these objects.

Keywords

groups, subgroups, residue classes, orders, cyclic groups, symmetric groups, homomorphisms of groups, normal subgroups, the fundamental theorem on group homomorphisms, conjugacy classes, class equations, actions of groups, solvable groups, representations of finite groups

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Definition of a group and examples Details will be announced during each lecture.
Class 2 Subgroups Details will be announced during each lecture.
Class 3 Order of an element of a group, cyclic groups Details will be announced during each lecture.
Class 4 Symmetric groups Details will be announced during each lecture.
Class 5 Right- and left-cosets by a subgroup Details will be announced during each lecture.
Class 6 Normal subgroups, residue groups Details will be announced during each lecture.
Class 7 Homomorphisms of groups, the fundamental theorems on group homomorphisms Details will be announced during each lecture.
Class 8 Actions of groups (1) Details will be announced during each lecture.
Class 9 Actions of groups (2) Details will be announced during each lecture.
Class 10 Sylow theorems Details will be announced during each lecture.
Class 11 Solvable groups Details will be announced during each lecture.
Class 12 Representations of finite groups (1) Details will be announced during each lecture.
Class 13 Representations of finite groups (2) Details will be announced during each lecture.
Class 14 Representations of finite groups (3) 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 evaluation of the results for midterm examination and final examination. Details will be announced during a lecture.

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.A204 ： Exercises in Algebra A 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.A204: Exercises in Algebra A II (if not passed yet) at the same time.

Other

None in particular.

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