Topics in Finite Group Theory
NGA Course
25 October - 30 November 2023
This is the website of the postgraduate course I am giving on Topics in Group Theory, jointly with Sesuai Yash Madanha, as part of the National Graduate Academy Coursework Initiative.
Course Description: Time permitting we will cover the following topics: Permutation groups, simplicity of A_n, groups of small order, permutation representations, p-groups, Sylow theorems, normal series, solvable and nilpotent groups, finite direct products, basis theorem, fundamental theorem of finite abelian groups, general linear group, some simple groups.
Prerequisites: Undergraduate degree in mathematics. Basic knowledge of groups, rings and fields and linear algebra.
Format: There will be 6 lectures, of 2 hours each, and a written test at the end. The test is only necessary for those doing the course for official credits, but others might want to write it too in order to cement their learning. Tutorial problems will also be given in class.
The first three weeks of lectures starting on 25 October will be delivered by Bernardo Rodrigues on Wednesdays and these will take place from 3:00 pm to 5:00 pm (SA time).
The last three weeks of the course will be delivered by Sesuai Yash Madanha and the lectures will be given on Mondays at a time that will be communicated accordingly.
Lectures will be online via zoom. The zoom link will be shared via e-mail. Lecture notes will be available on a shared drive whose link will be made available accordingly.
Lectures: The lecturers will provide an overview of the material, do some basic examples, and explain important concepts. In order to master the material, you will have to read the course notes on your own and do practice problems, again on your own.
Textbook
Our reference texts will be:
P.J. Cameron, Permutation groups, London Math. Soc. Student Texts, vol. 45, CUP, 1999.
J.D. Dixon and B. Mortimer, Permutation groups, GTM, vol. 163, Springer, 1996.
D. Dummit and R. Foote, Abstract Algebra, 3rd ed., Wiley, 2004.
D. J. S. Robinson, A Course in the Theory of Groups. New York: Springer-Verlag New York, Inc., 1996. GTM, 80.
John S. Rose, A Course on Group Theory (Dover Publications, New York, 1994).
J. J. Rotman, An Introduction to the Theory of Groups, vol. 148 of GTM, Springer-Verlag, 1994.