Introduction to Group Theory (MAT204) | GSÜ

Content

Objective

To introduce some basic algebraic structures (groups and rings) and how to study them

Course Content

Groups as symmetry measuring constructs, subgroups, normal subgroups, quotient groups, group homomorphisms, isomorphism theorems, group actions
Rings, subrings and ideals, isomorphism theorems, irreducible and prime elements

Course Learning Outcomes

To understand basic algebraic structures and to be able to put them to use

Teaching and Learning Methods

Inquiry based learning
Weekly exercice sessions

References

Mathématique L3 Algèbre, Aviva Szpirglas

Abstract Algebra: Theory and Applications, Thomas W. Judson, Robert A. Beezer
http://abstract.ups.edu/aata/aata.html

An Inquiry Based Approach to Abstract Algebra, Dana C. Ernst
https://danaernst.com/teaching/mat411f20/IBL-AbstractAlgebra.pdf

Cebir I - Temel Grup Teorisi, Ali Nesin
https://nesinkoyleri.org/wp-content/uploads/2019/05/cebir.pdf

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Notion of symmetry
2	Axiomatic definition of a group, group examples, operation tables, subgroup
3	Group homomorphisms, operations on groups
4	Kernel et image of homomorphisms, quotient of a group by a subgroup, Lagrange's theorem
5	Normal subgroups, quotient groups, Isomorphism theorems
6	Semi-direct products
7	Group actions on sets
8	Midterm
9	Orbit-stabiliser theorem, Sylow Theorems
10	Sylow Theorems and applications
11	Rings, ring homomorphisms, kernel and image of homomorphisms, subrings and ideals
12	Quotient rings, isomorphism theorem
13	Prime and irreducible elements
14	Unique Factorization Domains

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	3	60
Contribution of final exam to overall grade	1	40
Total	4	100

In-Term Studies
Activities	Number	Contribution
Assignments	0	0
Presentation	0	0
Midterm Examinations (including preparation)	1	40
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	2	20
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Make-up	0	0
Total	3	60

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1	understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways;					X
2	can properly state and use concepts and results of major mathematical interest;					X
3	masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools;	X
4	is able to express one’s mathematical ideas in an organised way both in written and oral forms;					X
5	understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...);					X
6	has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems;		X
7	has a theoretical and practical knowledge in computer science well adapted for learning a programming language;				X
8	has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions;				X
9	has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement;	X
10	masters French language as well as other foreign languages, to a level sufficient to study or work abroad.				X

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	5	70
Working Hours out of Class	14	3	42
Assignments	14	4	56
Presentation	4	3	12
Midterm Examinations (including preparation)	1	6	6
Project	0	0	0
Laboratory	0	0	0
Other Applications	2	2	4
Final Examinations (including preparation)	1	7	7
Quiz	2	4	8
Term Paper/ Project	0	0	0
Portfolio Study	0	0	0
Reports	0	0	0
Learning Diary	0	0	0
Thesis/ Project	0	0	0
Seminar	0	0	0
Other	0	0	0
Make-up	0	0	0
Yıl Sonu	0	0	0
Hazırlık Yıl Sonu	0	0	0
Hazırlık Bütünleme	0	0	0
Total Workload			205
Total Workload / 25			8.20
Credits ECTS			8