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2018　Introduction to Algebra III

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Naito Satoshi  Somekawa Mutsuro 
Class Format
Lecture / Exercise     
Media-enhanced courses
Day/Period(Room No.)
Fri3-8(H112)  
Group
-
Course number
MTH.A203
Credits
2
Academic year
2018
Offered quarter
3Q
Syllabus updated
2018/3/20
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 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.

Student learning outcomes

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.

Keywords

group, subgroup, residue class, order, cyclic group, symmetric group

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 accompanied by discussion sesssions.

Course schedule/Required learning

  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.
Class 15 Checking session Details will be provided during each class session.

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 discussion session and final examination. Details will be announced during a lecture.

Related courses

  • MTH.A201 ： Introduction to Algebra I
  • MTH.A202 ： Introduction to Algebra II
  • MTH.A204 ： Introduction to Algebra IV

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] and [Introduction to Algebra II].

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