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 will be succeeded by ``Introduction to Algebra IV'' in the fourth quarter.
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, and symmetric groups.
To become able to prove by him/herself basic properties of these objects.
group, subgroup, residue class, order, cyclic group, symmetric 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 | Axiom of groups, typical examples of groups, first properties of groups | Details will be provided during each class session. |
Class 2 | Discussion session on the axiom of groups, typical examples of groups, first properties of groups | Details will be provided during each class session. |
Class 3 | Basic properties of the operation in a group and of the identity and inverse elements | Details will be provided during each class session. |
Class 4 | Discussion session on basic properties of the operation in a group and of the identity and inverse elements | Details will be provided during each class session. |
Class 5 | Definition of a subgroup, criterion for subgroups, and examples of subgroups | Details will be provided during each class session. |
Class 6 | Discussion session on the definition of a subgroup, criterion for subgroups, and examples of subgroups | Details will be provided during each class session. |
Class 7 | Right- and left-cosets by a subgroup | Details will be provided during each class session. |
Class 8 | Discussion session on right- and left-cosets by a subgroup | Details will be provided during each class session. |
Class 9 | Order of a group, Lagrange's theorem | Details will be provided during each class session. |
Class 10 | Discussion session on the order of a group and Lagrange's theorem | Details will be provided during each class session. |
Class 11 | Order of an element of a group, cyclic groups | Details will be provided during each class session. |
Class 12 | Discussion session on the order of an element of a group and on cyclic groups | Details will be provided during each class session. |
Class 13 | Symmetric groups | Details will be provided during each class session. |
Class 14 | Discussion session on symmetric groups | Details will be provided during each class session. |
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 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] and [Introduction to Algebra II].
None in particular.