Matematica | GROUP THEORY

cod. 0522200019

GROUP THEORY

	0522200019
	DIPARTIMENTO DI MATEMATICA
	EQF7
	MATHEMATICS
	2022/2023

CURRICULUM GENERAL MATHEMATICS AND DIDACTICS OF MATHEMATICS

	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2018
	AUTUMN SEMESTER

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	MAT/02	6	48	LESSONS	SUPPLEMENTARY COMPULSORY SUBJECTS

EXAMS

	Exam	Date	Session
	TEORIA DEI GRUPPI	13/06/2023 - 12:00	SESSIONE ORDINARIA
	TEORIA DEI GRUPPI	11/07/2023 - 12:00	SESSIONE ORDINARIA

	Objectives
	THE AIM OF THIS COURSE IS TO INTRODUCE THE STUDENT TO THE GENERAL THEORY OF GROUPS. WE WILL CONSTRUCT MANY EXAMPLES OF GROUPS AND WE WILL DEVELOP THE BASIC PROPERTIES OF SOME CLASSES OF GROUPS. WE WILL ALSO PRESENT RECENT RESULTS IN THIS AREA.

	Prerequisites
	GOOD KNOWLEDGE OF THE SUBJECTS CONTAINED IN THE COURSES OF ALGEBRA I, ALGEBRA II.

	Contents
	FUNDAMENTAL CONCEPTS OF GROUP THEORY. PERMUTATION GROUPS. GROUP ACTIONS AND APPLICATIONS. SYLOW'S THEOREM. SERIES AND COMPOSITION SERIES. THE THEOREMS OF SCHREIER AND OF JORDAN-HOLDER. ABELIAN GROUPS: FREE ABELIAN GROUPS, DIVISIBLE GROUPS, FINITELY GENERATED ABELIAN GROUPS. SOLUBLE AND NILPOTENT GROUPS. FITTING SUBGROUP. FINITE SOLUBLE GROUPS: HALL'S THEOREMS. SUPERSOLUBLE GROUPS. SPLITTING THEOREMS. SCHUR-ZASSENHAUS THEOREM. FINITENESS CONDITIONS. POLYCYCLIC GROUPS. CERNIKOV GROUPS.

FUNDAMENTAL CONCEPTS OF GROUP THEORY. PERMUTATION GROUPS.
GROUP ACTIONS AND APPLICATIONS. SYLOW'S THEOREM.
SERIES AND COMPOSITION SERIES. THE THEOREMS OF SCHREIER AND OF JORDAN-HOLDER.
ABELIAN GROUPS: FREE ABELIAN GROUPS, DIVISIBLE GROUPS, FINITELY GENERATED ABELIAN GROUPS.
SOLUBLE AND NILPOTENT GROUPS. FITTING SUBGROUP.
FINITE SOLUBLE GROUPS: HALL'S THEOREMS. SUPERSOLUBLE GROUPS.
SPLITTING THEOREMS. SCHUR-ZASSENHAUS THEOREM.
FINITENESS CONDITIONS. POLYCYCLIC GROUPS. CERNIKOV GROUPS.

	Teaching Methods
	LECTURES. ATTENDANCE TO CLASS LESSONS IS STRONGLY RECOMMENDED.

	Verification of learning
	THE AIM OF THE EXAMINATION IS TO EVALUATE THE FAMILIARITY OF THE STUDENT WITH THE THEORY OF GROUPS. THE EXAMINATION IS ORAL. THE STUDENT HAS TO TALK ABOUT EXAMPLES AND THE PRINCIPAL PROPERTIES OF SOME CLASSES OF GROUPS AND HAS TO SOLVE SOME EXERCISES. OPTIONAL IS TO PRESENT A SHORT SEMINAR ON A TOPIC NOT PRESENTED IN THE LECTURES.

	Texts
	D.J.S. ROBINSON, A COURSE IN THE THEORY OF GROUPS, SPRINGER VERLAG, 1996. J.S. ROSE, A COURSE ON GROUP THEORY, DOVER, 1994. M. CURZIO, P. LONGOBARDI, M. MAJ - LEZIONI DI ALGEBRA - LIGUORI EDITORE, NAPOLI, II EDIZIONE 2014 D.J.S. ROBINSON, AN INTRODUCTION TO ABSTRACT ALGEBRA, DE GRUYTER, 2004.