COURSE UNIT TITLE

: ABELIAN GROUPS

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 5061 ABELIAN GROUPS ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

ASSOCIATE PROFESSOR ENGIN MERMUT

Offered to

Mathematics
Mathematics

Course Objective

The aim of this course is to introduce the main techniques and the main structure results in abelian group theory.

Learning Outcomes of the Course Unit

1   Will be able to use the powerful structure theorems in abelian group theory.
2   Will be able to understand the projective and injective objects in the category of abelian groups as free and divisible abelian groups whose structure are completely determined.
3   Will be able to use pure subgroups and basic subgroups when working on problems in abelian groups.
4   Will be able to understand how algebraically compact groups turn out to be pure-injective groups and how the structure of algebraically compact groups is determined.
5   Will be able to understand the structure of character groups and the duality between discrete torsion and totally disconnected compact groups.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Basic notions on abelian groups. Torsion groups. Modules. Categories of abelian groups.
2 Direct sums and direct products. Direct summands. Pullback and pushout. Direct and inverse limits.
3 Free abelian groups. Finitely generated groups.
4 Divisible groups. Injective groups. The divisible hull. Finitely cogenerated groups. Structure theorem for divisible groups.
5 Linear independence and rank. Subgroups of direct sums of cyclic groups. Countable free groups. Rank-one torsion-free groups. Structure of completely decomposable groups.
6 Pure subgroups. Groups of bounded order. Height. Quotient groups modulo pure subgroups. Pure-exact sequences. Pure-projectivity and pure-injectivity.
7 Generalization of purity. Neat subgroups.
8 Midterm
9 Basic subgroups. The Ulm sequence. The structure of countable torsion groups. Functorial subgroups and quotient groups.
10 Topologies in groups. Topological groups and linear topologies. Completeness and completions.
11 The algebraic structure of compact abelian groups. Algebraically compact groups. Structure of algebraically compact groups. Pure-essential extensions.
12 Valuation rings. Torsion-free modules. Complete modules. Characteristic submodules. The ring of endomorphisms.
13 Homomorphism groups. Exact sequences for Hom. Character groups. Duality between discrete torsion and 0-dimensional compact groups.
14 p-groups. Torsion-free groups. Mixed groups. Subgroup lattices of groups.

Recomended or Required Reading

Textbook(s):
[1] Laszlo Fuchs. Infinite abelian groups. Vol. I. Academic Press, New York, 1970.
[2] Laszlo Fuchs. Infinite abelian groups. Vol. II. Academic Press, New York, 1973.
[3] Irving Kaplansky. Infinite abelian groups. Revised edition. The University of Michigan Press, Ann Arbor, Mich., 1969.
[4] Grigore Calugareanu, Simion Breaz, Ciprian Modoi, Cosmin Pelea, and Dumitru Valcan. Exercises in abelian group theory, volume 25 of Kluwer Texts in the Mathematical Sciences. Kluwer Academic Publishers Group, Dordrecht, 2003.

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 ASG ASSIGNMENT
2 MTE MIDTERM EXAM
3 FIN FINAL EXAM
4 FCG FINAL COURSE GRADE ASG * 0.30 + MTE * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) ASG * 0.30 + MTE * 0.30 + RST * 0.40


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

Attending at least 70 percent of lectures is mandatory.

Contact Details for the Lecturer(s)

e-mail: engin.mermut@deu.edu.tr
Office: (232) 301 85 82

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Lectures 13 3 39
Preparations before/after weekly lectures 13 3 39
Preparation for midterm exam 1 15 15
Preparation for final exam 1 25 25
Preparing assignments 10 5 50
Final 1 3 3
Midterm 1 3 3
TOTAL WORKLOAD (hours) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11
LO.143222433
LO.243222433
LO.343222433
LO.443222433
LO.543222433

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