2018 Introduction to Algebra II

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Academic unit or major
Mathematics
Instructor(s)
Naito Satoshi 
Course component(s)
Lecture
Day/Period(Room No.)
Fri3-4(H112)  
Group
-
Course number
ZUA.A203
Credits
2
Academic year
2018
Offered quarter
3-4Q
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 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.

Student learning outcomes

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.

Keywords

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

Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
- - - -

Class flow

Standard lecture course

Course schedule/Required learning

  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 evaluation of progress Details will be announced during each lecture.
Class 9 Homomorphisms of groups, image and kernel of a homomorphism of groups Details will be announced during each lecture.
Class 10 Normal subgroups, residue groups Details will be announced during each lecture.
Class 11 The first, second and third fundamental theorems on group homomorphisms Details will be announced during each lecture.
Class 12 Subgroups generated by subsets Details will be announced during each lecture.
Class 13 Conjugacy of elements, conjugacy classes, centralizers Details will be announced during each lecture.
Class 14 Class equation and its applications Details will be announced during each lecture.
Class 15 Actions of groups Details will be announced during each lecture.

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

Midterm exam and final exam. Details will be announced during a lecture.

Related courses

  • MTH.A203 : Introduction to Algebra III
  • MTH.A204 : Introduction to Algebra IV
  • ZUA.A201 : Introduction to Algebra I
  • ZUA.A202 : Exercises in Algebra A I
  • ZUA.A204 : Exercises in Algebra A II

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 (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.

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