Introduction to Algebra II

Academic unit or major

Mathematics

Instructor(s)

Kazuma Shimomoto

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

3-4 Fri

Class

-

Course Code

ZUA.A203

Number of credits

200

Course offered

2024

Offered quarter

3-4Q

Syllabus updated

Mar 17, 2025

Language

Japanese

Syllabus

Course overview and goals

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.

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

Standard lecture course.

Course schedule/Objectives

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 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

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.