ADVANCED ALGEBRA B
- Overview
- Assessment methods
- Learning objectives
- Contents
- Bibliography
- Teaching methods
- Contacts/Info
Knowledge of basic results in group theory: subgroups, homomorphisms, conjugation classes, Lagrange's theorem, isomorphism theorems, direct and semi-direct products, Sylow's theorem, classification of finitely generated abelian groups.
Written examination of 2 hours. The written examination consists of 3 or 4 exercises which will require the application of some of the algorithms seen in class and the ability of independently prove some results relying on the properties presented in class.
The oral proof, starting immediately, usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class and to apply them to concrete situations. It will be object of evaluation the ability to present a proof in a complete and rigorous
Passing examination and the final grading depend both on oral and written tests.
Knowledge od some topics in group theory: free groups and presentations of groups and nilpotent groups.
At the end of the course the student will be able to study the properties of a group that can be derived from its presentation. Moreover he will use commutator calculus to analyze a group.
Congruences in monoids and groups. Relations and relators.
Free monoids and groups. Definition and elementary properties. Existence and unicity. Nielsen-Schreier theorem.
Presentations of groups. Tietze transformations. Todd-Coxeter method.
Nilpotent groups. Upper and lower central series. The normalizer condition. Finite nilpotent groups. The Hall collection formula. The Lie algebra of a nilpotent group.
D. J.S. Robinson, A Course in the Theory of Groups, Springer
D. L. Johnson, Presentations of groups, LMS
D. E. Cohen, Combinatorial Group Theory
a topological approach, LMS
A. E. Clement, S. Majewicz, M. Zyman, The Theory of Nilpotent Groups, Birkhauser
Frontal lectures: 64 hours
For further detail go to the web page of the course.