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	
MA262	Abelian Groups (1)	

Semester	Hours: C+S+L	Category	Status	
1	2+2+1		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
COVACI Rodica	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 general theory of groups and elements of the theory of group representations.	

IV. Course contents
1. Rudiments of group theory: group, subgroup, normal subgroup, homomorphisms, the isomorphism theorems, permutation groups.
2. Automorphisms: the automorphism group of a group, inner automorphisms, characteristic subgroups, the derived group of a group.
3. Group actions, representations by permutations: action of a group on a set, the associated representation, faithful actions, orbits and stabilizers, transitive actions, the orbit-stabilizer theorem, some applications.
4. Local structure of finite groups: Sylow’s theorems, Cauchy’s theorem, applications.
5. Normal structure of groups: composition series, solvable groups, nilpotent groups.	

V. Bibliography
1. ALPERIN, J.L.; BELL, R.B., Groups and representations, Springer-Verlag, New York, 1995. 
2. BECHEANU, M., etc., Algebra, Editura ALL, Bucuresti, 1998.
3. CRIVEI, S., Basic abstract algebra, Casa Cartii de Stiinta, Cluj-Napoca, 2002.
4. HUPPERT, B., Endliche Gruppen I, Springer-Verlag, Berlin - New York, 1967.
5. PURDEA, I.; POP, I., Algebra, Editura GIL, Zalau, 2003.
6. ROTMAN, J., An Introduction to the Theory of Groups, Springer-Verlag, New York, 1995.

VI. Thematic of didactic activities per weeks
Week 1.  Rudiments of group theory. Group, subgroup, cyclic group, index, Lagrange’s theorem, normal subgroup, quotient group (factor group).
      [1, Chap.1, 1.]; [5, Cap.II, 2.3, 2.5, 2.6, 2.7, 2.8]; [3, Chap.2, 2.4, 2.5, 2.7, 2.8]
Week 2.  Homomorphisms, Correspondence theorem, Isomorphism theorems. Direct product of groups.
      [1, Chap.1, 1.]; [3, Chap.2, 2.6, 2.9]; [5, Cap.II, 2.4, 2.9, 2.10, 2.16]
Week 3.  The automorphism group of a group. Inner automorphisms. Characteristic subgroups, center of a group, commutators, the derived group.
      [1, Chap.1, 2.]; [5, Cap.II, 2.12]
Week 4.  Permutation groups. The symmetric group of a set, the symmetric group and the alternating group of degree n, cycles, disjoint cycle decomposition of a permutation.
      [1, Chap.1, 1.]; [2, Cap.IV, 8.]; [5, Cap.II, 2.11]
Week 5.  Action of a group on a set, the associated representation by permutations. Faithful actions, Cayley’s theorem (as a representation theorem).
      [1, Chap.1, 3.]
Week 6.  Orbits and stabilizers. Transitive actions. G-set homomorphisms, isomorphic G-sets.
      [1, Chap.1, 3.]
Week 7.  The orbit-stabilizer theorem. Some applications: the centralizer and the cardinality of a conjugacy class of a finite group; the normalizer and the number of conjugates of a subgroup in a finite group.
      [1, Chap.1, 3.]
Week 8.  Local structure of finite groups. Sylow’s theorems and consequences. Cauchy’s theorem.
      [1, Chap.3, 7.]; [5, Cap.II, 2.14, 2.15]
Week 9.  Some properties and applications of Sylow subgroups.
      [1, Chap.3, 7., 8.]; [5, Cap.II, 2.15]
Week 10. Normal structure of groups. Descending series of subgroups. Composition series.
      [1, Chap.4, 10.]
Week 11. Derived series. Solvable groups.
      [1, Chap.4, 11.]; [5, Cap.II, 2.13]
Week 12. Hall subgroups in finite solvable groups.
      [4, Kap.VI, 1.]
Week 13. Nilpotent groups.
      [1, Chap.4, 11.]
Week 14. Carter subgroups in finite solvable groups. Rudiments of formation theory.
      [4, Kap.VI, 7.]                                                                                               	

VII. Didactic methods used
Exposition, lecture, conversation, didactic proof, problem-questions, individual study, exercise, reports.	

VIII. Assessment
Report(50%) + Exam(50%).	

IX. Additional references
1.	Popescu, D.; Vraciu, C., Elemente de teoria grupurilor finite, Editura Stiintifica si Enciclopedica, Bucuresti, 1986.
2.	Purdea, I.; Pic, Gh., Tratat de algebra moderna, vol.I, Editura Academiei, Bucuresti, 1977.
3.	Purdea, I.; Pelea, C., Probleme de algebra, Editura EIKON, Cluj-Napoca, 2008.	

	Dean,							Chair head,
            Prof. TAMBULEA Leon, Ph.D.                              Assoc.Prof. BREAZ Simion, Ph.D.

	Date, 							Course responsible,
	3.10.2008						Assoc.Prof. COVACI Rodica, Ph.D.