Algebra is a discipline of mathematics that deals with abstract notions which generalize algebraic operations on various mathematical objects. The main subjects of of this course include basic notions and properties of groups, which are a mathematical object having just one operation. To help deeper understanding of the newly learnt concepts, each even-numbered class is devoted to a discussion session, where excercises are given related to the contents of the preceding lecture. This course succeeds ``Introduction to Algebra III'' offered in the third quater.
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 of groups, 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 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.
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 accompanied by discussion sesssions.
Course schedule | Required learning | |
---|---|---|
Class 1 | Homomorphisms of groups, image and kernel of a homomorphism of groups | Details will be provided during each class session. |
Class 2 | Discussion session on homomorphisms of groups and image and kernel of a homomorphism of groups | Details will be provided during each class session. |
Class 3 | Normal subgroups, residue groups | Details will be provided during each class session. |
Class 4 | Discussion session on normal subgroups and residue groups | Details will be provided during each class session. |
Class 5 | The first, second and third fundamental theorems on group homomorphisms | Details will be provided during each class session. |
Class 6 | Discussion session on the first, second and third fundamental theorems on group homomorphisms | Details will be provided during each class session. |
Class 7 | Subgroups generated by subsets | Details will be provided during each class session. |
Class 8 | Discussion session on subgroups generated by subsets | Details will be provided during each class session. |
Class 9 | Conjugacy of elements, conjugacy classes, centralizers | Details will be provided during each class session. |
Class 10 | Discussion session on conjugacy of elements, conjugacy classes, and centralizers | Details will be provided during each class session. |
Class 11 | Class equation and its applications | Details will be provided during each class session. |
Class 12 | Discussion session on the class equation and its applications | Details will be provided during each class session. |
Class 13 | Actions of groups | Details will be provided during each class session. |
Class 14 | Discussion session on actions of groups | Details will be provided during each class session. |
Class 15 | Checking session | Details will be provided during each class session. |
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 discussion session 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], [Introduction to Algebra II] and [Introduction to Algebra III].