Print

2017　Introduction to Algebra II

Font size  SML

Register update notification mail Add to favorite lecture list
Academic unit or major
Mathematics
Instructor(s)
Taguchi Yuichiro 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Fri3-4(H112)  
Group
-
Course number
ZUA.A203
Credits
2
Academic year
2017
Offered quarter
3-4Q
Syllabus updated
2017/3/17
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

✔ Specialist skills Intercultural skills Communication 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
Class 3 Definition of a subgroup, criterion for subgroups, and examples of subgroups
Class 4 Right- and left-cosets by a subgroup
Class 5 Order of a group, Lagrange's theorem
Class 6 Order of an element of a group, cyclic groups
Class 7 Symmetric groups
Class 8 evaluation of progress
Class 9 Homomorphisms of groups, image and kernel of a homomorphism of groups
Class 10 Normal subgroups, residue groups
Class 11 The first, second and third fundamental theorems on group homomorphisms
Class 12 Subgroups generated by subsets
Class 13 Conjugacy of elements, conjugacy classes, centralizers
Class 14 Class equation and its applications
Class 15 Actions of groups

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

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.

Page Top