FURTHER TOPICS IN ALGEBRA
Academic year and teacher
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- Versione italiana
- Academic year
- 2016/2017
- Teacher
- GIULIANO BIANCO
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/02
Training objectives
- The lectures are about the Theory of Finite Groups, with special attention to the tools useful for their classification
Prerequisites
- Algebra.
Course programme
- Sylow Theory (12h)
Group actions, Sylow Theorems, p-Groups and Nilpotent Groups, non-semplicity criteria based on the group cardinality, abelian Sylow subgroups, normal abelian subgroups.
Subnormality (8h)
The lattice of subnormal subgroups, Zipper Lemma and applications, locality and p-locality, abelian and cyclic subgroups.
Split Extensios (12h)
Normal subgroups complements and split extensions, Schur-Zassenhaus Theorem, solvable groups, Hall subgroups and solvability, \pi-separability, coprime actions and fusion, cyclic extensions.
Commutator Theory (8h)
Commutator subgroups and central series, the three subgroups Lemma and applications, subgroups which admit an automorphism action, coprime automorphism actions.
Transfer Theory (8h)
The Transfer morphism and its properties, the tranfer evaluation Lemma, the tansfer kernel and fusion, tranfer aimed at a non abelian Sylow subgroup, conditions for the existence of normal p-complements. Didactic methods
- Lectures.
Learning assessment procedures
- Oral examination. There is also the possibility for the students to prepare an exposition on a subject of their choice.
Reference texts
- For following the course the notes by the students
themselves are sufficient. The main arguments are
also contained in the initial chapters of the
following books which may be used mainly for
more advanced study.
A. Arhangelskii/M. Tkachenko: Topological groups and
related structures. World Scientific 2008.
E. Brigham: The fast Fourier transform and its applications. Prentice-Hall 1988.
E. Hewitt/K. Ross: Abstract harmonic analysis I. Springer 1979.
H. Reiter/J. Stegeman: Classical harmonic analysis and
locally compact groups. Oxford UP 2000.
M. Stroppel: Locally compact groups. EMS 2006.
S. Willard: General topology. Dover 2004.
