Introduction to Group Theory

6 ECTS credits
180 h study time

Offer 1 with catalog number 1007406ANR for all students in the 2nd semester at a (A) Bachelor - preliminary level.

Semester

2nd semester

Enrollment based on exam contract

Impossible

Grading method

Grading (scale from 0 to 20)

Can retake in second session

Yes

Enrollment Requirements

Students who want to enroll for this course, must have passed or be enrolled for "Linear Algebra".

Taught in

Dutch

Faculty

Faculty of Sciences and Bioengineering Sciences

Department

Mathematics

Educational team

Arne Van Antwerpen (course titular)

Activities and contact hours

26 contact hours Lecture
26 contact hours Seminar, Exercises or Practicals

Course Content

Although there are many applications of algebra in other disciplines, such as physics, chemistry and computer sciences, this course is restricted to ``abstract algebra''. We study algebraic systems, in particular, we investigate groups.

We give an introduction to group theory. Via this study one learns also how to reason in and understand abstract theories, and one learns strategies for solving some problems.

The following topics will be covered: groups of small order, permutation groups, linear groups, cyclic groups, group generators, normal subgroups,

quotient groups, homomorphism theorems, Lagrange's and Cauchy's theorem, the fundamental theorem of finite

and finitely generated abelian groups, actions of groups and the Sylow theorems.

The theory will be illustrated with many examples. During the tutorials more examples will be investigated and also some important techniques in group theory will be dealt with. The aim is to develop an algebraic intuition and to learn several methods

for proving results.

Table of Contents (every week 2 hours of class)

1. Introductions

1.1 Sets

1.2 Some well known results from number theory

1.3 Functions

2. Definition and Examples

2.1 Definitions

2.2 Examples

2.3 Rings and more examples

2.4 Multiplication table

2.5 Eelementary Properties

2.6 The order of an element

2.7 Equations in groups

2.8 Direct products

3. Subgroups

3.1 Definition

3.2. Special subgroups

3.3 Generators

4. Cosets

4.1 Definition

4.2 Theorem of Lagrange

4.3 Applications

5. Normal Subgroups

6. Quotient groups

6.1 Definition

6.2 Subgroups and quotient groups

7. Homomorphism

7.1 Definition

7.2 Isomorphism

7.3 Homomorphism Theorems

8. Permutation groups

8.1 Theorem of Cayley

8.2 Finite Permutation Groups

9. Finite Abelian Groups

9.1 Direct Products

9.2 Fundamental Theorem

10. Actions

10.1 Definition

10.2 Orbit-Stabilers Theorem

10.3 Semidirect products of groups

10.4 Sylow Theorems

11. Excersises

Exercises sessions: each week 2 hours.

Course material

Digital course material (Required) : Algebra 1
Digital course material (Required) : Nota's beschikbaar, website: http://homepages.vub.ac.be/~efjesper
Handbook (Recommended) : Algebra, M. Artin, 2de, Pearson, 9781292027661, 2013
Handbook (Recommended) : Abstract algebra, I.N. Herstein, Wiley, 9780471368793, 1996
Handbook (Recommended) : A course in group theory, J.F. Humphreys, Oxford Science Publications, Oxford, 9780198534594, 1996

Additional info

Exercise sessions consist of exercises in class and independent home works. During the class sessions all types of representative exercises will be solved with the aid of the instructor. Regularly home works will be assigned that are somehow related to the type of exercises solved in class. Every student must submit their solutions, in writing, to the assistant, and this before the given deadline. This document will be evaluated on the mathematical content as well as on the writing style and lay out. Course notes available on the web: http://homepages.vub.ac.be/~efjesper

Complementary study material:

M. Artin, Algebra, Prentice Hall, London, 1991. (ISBN: 0-13-004763-5)

I.N. Herstein, Abstract algebra, Prentice Hall, 1996.

J.F. Humphreys, A course in group theory, Oxford Science

Publications, Oxford, 1996.

Learning Outcomes

General competencies

1. Student knows basic concepts of introductory group theory.

2. Student can apply the basic concepts of group theory on examples.

3. Student can recognize other basic concepts of group theory in examples and proofs of properties.

4. Student can think in function of problem solving.

5. Student can reconstruct simple proofs.

6. Student can write a mathematical text independently about solutions of exercises.

7. Student has knowledge of software package GAP, Groups Algebras and Programming.

Grading

The final grade is composed based on the following categories:
Other Exam determines 100% of the final mark.

Within the Other Exam category, the following assignments need to be completed:

examen with a relative weight of 1 which comprises 100% of the final mark.

Additional info regarding evaluation

Written Exam on Exercises (40%) and Oral Exam (theory,

60%). A mark will only be assigned if the student has participated in all test and exams.

Allowed unsatisfactory mark

The supplementary Teaching and Examination Regulations of your faculty stipulate whether an allowed unsatisfactory mark for this programme unit is permitted.

Academic context