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2024　Introduction to Algebra II

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Academic unit or major
Mathematics
Instructor(s)
Shimomoto Kazuma 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
-
Group
-
Course number
ZUA.A203
Credits
2
Academic year
2024
Offered quarter
3-4Q
Syllabus updated
2024/3/14
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 able to explain important notions such as groups, subgroups, orders, cyclic groups, symmetric groups, residue classes, normal subgroups, homomorphisms of groups, the fundamental theorem on group homomorphisms, actions of groups, orbits, conjugacy classes, class equations, Sylow theorems, solvable groups, representations of finite groups, and character of representations, together with their examples.

To become able to prove basic properties of these objects by him/herself.

Keywords

groups, subgroups, orders, cyclic groups, symmetric groups, residue classes, normal subgroups, homomorphisms of groups, the fundamental theorem on group homomorphisms, actions of groups, orbits, conjugacy classes, class equations, Sylow theorems, solvable groups, representations of finite groups, character of representations

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 Definition of a group and examples Details will be announced during each lecture.
Class 2 Subgroups Details will be announced during each lecture.
Class 3 Order of an element of a group, cyclic groups Details will be announced during each lecture.
Class 4 Symmetric groups Details will be announced during each lecture.
Class 5 Right- and left-cosets by a subgroup Details will be announced during each lecture.
Class 6 Normal subgroups, quotient groups Details will be announced during each lecture.
Class 7 Homomorphisms of groups, the fundamental theorems on group homomorphisms Details will be announced during each lecture.
Class 8 Definition of actions of groups and their examples, stabilizers, orbits, orbit decompositions Details will be announced during each lecture.
Class 9 Conjugate by an element of a group, conjugacy classes, class equations Details will be announced during each lecture.
Class 10 Application of actions of groups, Sylow theorems Details will be announced during each lecture.
Class 11 Solvable groups Details will be announced during each lecture.
Class 12 Definition of representations of finite groups and their examples Details will be announced during each lecture.
Class 13 Schur's lemma, Maschke's theorem Details will be announced during each lecture.
Class 14 Character of representations Details will be announced during each lecture.

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

None.

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 midterm examination and final examination. 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.

Other

None in particular.

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