Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Algebră şi Geometrie (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MA263	Abelian Groups (2)	

Semester	Hours: C+S+L	Category	Status	
2	2+2+0		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
BREAZ Simion Sorin	Ph.D.	Assoc.Prof.	Algebra	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
Basic notions and results concerning divisibility, purity, basic subgroups and Ulm-Kaplansky invariants for abelian groups.	

IV. Course contents
Basic knowledges concerning divisibility (equivalence to injectivity, structure theorem), purity (the notion of purity in lattices, characterizations of purity for subgroups), basic subgroups (existence and uniqueness up to an isomorphism), Ulm-Kaplansky invariants and Ulm's theorem.	

V. Bibliography
1. L. FUCHS: Infinite Abelian Groups, vol.1, Academic Press, 1970. 
2. G. CALUGAREANU: Introducere laticiala in teoria grupurilor abeliene, Editura Expert, 1994.
3. G. CALUGAREANU, S. BREAZ, C. MODOI, C. PELEA, D. VALCAN: Exercises in abelian group theory, Kluwer Academic Publishers Group, Dordrecht, 2003
4. J.J. ROTMAN: An introduction to the theory of groups. Fourth edition.  Springer-Verlag, New York, 1995.
5. S. BREAZ, G. GALUGAREANU, Fundamentele teoriei grupurilor abeliene, Editura Academiei Romane, 2005.

VI. Thematic of didactic activities per weeks
1.	Preliminaries. (1 lecture – Week 1 + 1 seminar – Week 1)
Keywords: Groups, Abelian groups, rings and modules.
2.	Free groups and examples of groups (1 lecture – Week 2 + 1 seminar – Week 2)
Keywords: Free groups, generators and relations.
3.	Special subgroups, cocyclic groups (1 lecture – Week 3 + 1 seminar – Week 3)
Keywords: subgroups, height, cocyclic groups, Generalized Pruffer group.
4.	Homomorphisms and diagrams, universal properties (2 lectures – Week 4, 5 + 2 seminars – Week 4, 5)
Keywords: commutative diagram, direct sum, pullback, pushout, exact sequences, fundamental lemmas.
5.	Independence and rank (1 lecture – Week 6 + 1 seminar – Week 6)
Keywords: essential subgroup, superfluous subgroups, linear independent system, rank.
6.	Divisible groups (1 lecture – Week 7 + 1 seminar – Week 7)
Keywords: Divisible group, injective groups, projective groups
7.	Finitely generated groups (1 lecture – Week 8 + 1 seminar – Week 8)
Keywords: Finitely generated groups, Fundamental Theorem, Sigma-groups
8.	Tensorial product and pure subgroups (2 lectures – Week 9, 10 + 2 seminars – Week 9, 10)
Keywords: Tensorial product, pure subgroup, pure exact sequence, pure injectivity, pure projectivity.
9.	p-groups (2 lectures – Week 11, 12 + 2 seminars – Week 11, 12)
Keywods: basic subgroup, Ulm-Kaplansky invariants, Ulm’s Theorem.
10.	Torsion-free groups (2 lectures – Week 13, 14 + 2 seminars – Week 13, 14)
Keywords: characteristic, type, rank 1 groups, completely decomposable groups, separable groups, finite torsion-free rank groups.

VII. Didactic methods used
Lectures, presentations, conversations, challenge, projects, exercises, individual study, homework assignments	

VIII. Assessment
A written final exam (grade E), a test at the seminar (grade T) and a referee (R). The exam subjects have theoretical questions from all the studied topics, and one problem, among the problems studied at the course and last 4 seminars. The test subject have practical questions (exercices and problems) from topics which are studied in first 10 weeks. The final grade is the weighted mean of the three grades mentioned above, conditioned by all the grades being at least 5 from 10. Otherwise, the exam will not be passed.
The final grade = 50%E + 25%T + 25%R.

IX. Additional references
1.Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules,
Graduate Texts in Math. Vol. 13, Springer-Verlag, 1992.

2.Lam, T.Y.: Lectures On Modules and Rings, Graduate Texts in
Math. Vol. 189, Springer-Verlag, 1999.

3.Lam, T.Y.: A First Course in Noncommutative rings, Graduate
Texts in Math. Vol. 131, Springer-Verlag, 1991.

4.Lam, T.Y.: Exercices in Classical Ring Theory, Problem Books in
Mathematics, Springer-Verlag, 1995.

5.Lam, T.Y.: Exercices in Modules and Rings, Problem Books in
Mathematics, Springer-Verlag, 2007.

6.Stenstrom, B.: Ring of Quotients, Graduate Texts in Math.,
Springer-Verlag, 1975.

7.Wickless, W.: A First Course in Graduate Algebra, Taylor and
Francis, 2004.

	Date,							Course responsible,
								Assoc.Prof. BREAZ Simion Sorin, Ph.D.