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2016　Introduction to Algebra III

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Academic unit or major
Undergraduate major in Mathematics
Instructor(s)
Taguchi Yuichiro  Yamakawa Daisuke  Kawachi Takeshi 
Class Format
Lecture / Exercise     
Media-enhanced courses
Day/Period(Room No.)
Fri3-8(H112)  
Group
-
Course number
MTH.A203
Credits
2
Academic year
2016
Offered quarter
3Q
Syllabus updated
2016/4/27
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
Class 3 Basic properties of the operation in a group and of the identity and inverse elements
Class 4 Discussion session on basic properties of the operation in a group and of the identity and inverse elements
Class 5 Definition of a subgroup, criterion for subgroups, and examples of subgroups
Class 6 Discussion session on the definition of a subgroup, criterion for subgroups, and examples of subgroups
Class 7 Right- and left-cosets by a subgroup
Class 8 Discussion session on right- and left-cosets by a subgroup
Class 9 Order of a group, Lagrange's theorem
Class 10 Discussion session on the order of a group and Lagrange's theorem
Class 11 Order of an element of a group, cyclic groups
Class 12 Discussion session on the order of an element of a group and on cyclic groups
Class 13 Symmetric groups
Class 14 Discussion session on symmetric groups
Class 15 Checking session

Textbook(s)

None in particular

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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