# Matematica | GROUP THEORY

## Matematica GROUP THEORY

 0522200019 DIPARTIMENTO DI MATEMATICA EQF7 MATHEMATICS 2022/2023

 YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/02 6 48 LESSONS SUPPLEMENTARY COMPULSORY SUBJECTS
 MERCEDE MAJ T
ExamDate
TEORIA DEI GRUPPI13/06/2023 - 12:00
TEORIA DEI GRUPPI11/07/2023 - 12:00
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.
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.