2016 Introduction to Algebra IV

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Academic unit or major
Undergraduate major in Mathematics
Taguchi Yuichiro  Yamakawa Daisuke  Kawachi Takeshi 
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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 succeeds ``Introduction to Algebra III'' offered in the third quater.

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 of groups, 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 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.


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 accompanied by discussion sesssions.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Homomorphisms of groups, image and kernel of a homomorphism of groups Details will be provided during each class session.
Class 2 Discussion session on homomorphisms of groups and image and kernel of a homomorphism of groups
Class 3 Normal subgroups, residue groups
Class 4 Discussion session on normal subgroups and residue groups
Class 5 The first, second and third fundamental theorems on group homomorphisms
Class 6 Discussion session on the first, second and third fundamental theorems on group homomorphisms
Class 7 Subgroups generated by subsets
Class 8 Discussion session on subgroups generated by subsets
Class 9 Conjugacy of elements, conjugacy classes, centralizers
Class 10 Discussion session on conjugacy of elements, conjugacy classes, and centralizers
Class 11 Class equation and its applications
Class 12 Discussion session on the class equation and its applications
Class 13 Actions of groups
Class 14 Discussion session on actions of groups
Class 15 Checking session


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.A203 : Introduction to Algebra III

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], [Introduction to Algebra II] and [Introduction to Algebra III].

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