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.
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.
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
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | Axiom of groups, typical examples of groups, first properties of groups | Details will be announced during each lecture. |
Class 2 | Basic properties of the operation in a group and of the identity and inverse elements | Details will be announced during each lecture. |
Class 3 | Definition of a subgroup, criterion for subgroups, and examples of subgroups | Details will be announced during each lecture. |
Class 4 | Right- and left-cosets by a subgroup | Details will be announced during each lecture. |
Class 5 | Order of a group, Lagrange's theorem | Details will be announced during each lecture. |
Class 6 | Order of an element of a group, cyclic groups | Details will be announced during each lecture. |
Class 7 | Symmetric groups | Details will be announced during each lecture. |
Class 8 | Homomorphisms of groups, image and kernel of a homomorphism of groups | Details will be announced during each lecture. |
Class 9 | Normal subgroups, residue groups | Details will be announced during each lecture. |
Class 10 | The first, second and third fundamental theorems on group homomorphisms | Details will be announced during each lecture. |
Class 11 | Subgroups generated by subsets | Details will be announced during each lecture. |
Class 12 | Conjugacy of elements, conjugacy classes, centralizers | Details will be announced during each lecture. |
Class 13 | Class equation and its applications | Details will be announced during each lecture. |
Class 14 | Actions of groups | Details will be announced during each lecture. |
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.
Shoichi Nakajima : Basics of Algebra and Arithmetic, Kyoritsu Shuppan. Co., Ltd., 2000.
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.
Based on evaluation of the results for midterm examination and final examination. Details will be announced during a lecture.
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.
None in particular.