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.
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.
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: https://github.com/vendramin/group
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.
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.
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:
One task will be given during the semester. This yields 20% of the final grade.
There is then a written exam that assesses understanding of both theory and applications through a selection of exercises. This written exam yields 80% of your final grade.
An absence on one of the two parts of the exam implies an absence as the final result. The grade for the task is retained if one takes part in the second examination period.
This offer is part of the following study plans:
Bachelor of Business Economics: Minor Minor Education (only offered in Dutch)
Bachelor of Physics and Astronomy: Default track (only offered in Dutch)
Bachelor of Political Science and Sociology: - afstudeerrichting sociologie, minor onderwijs (only offered in Dutch)
Bachelor of Mathematics and Data Science: Standaard traject (only offered in Dutch)
Master of Teaching in Social Sciences: politieke wetenschappen en sociologie (90 ECTS, Etterbeek) (only offered in Dutch)
Master of Teaching in Science and Technology: biologie (120 ECTS, Etterbeek) (only offered in Dutch)
Master of Teaching in Science and Technology: geografie (120 ECTS, Etterbeek) (only offered in Dutch)
Master of Teaching in Science and Technology: chemie (120 ECTS, Etterbeek) (only offered in Dutch)
Master of Teaching in Science and Technology: fysica (120 ECTS, Etterbeek) (only offered in Dutch)
Master of Teaching in Science and Technology: computerwetenschappen (120 ECTS, Etterbeek) (only offered in Dutch)
Master of Teaching in Science and Technology: ingenieurswetenschappen (120 ECTS, Etterbeek) (only offered in Dutch)
Master of Teaching in Economics: standaard traject (90 ECTS, Etterbeek) (only offered in Dutch)