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MATH 529

Finite Group Theory

MATH 529 is a graduate deep-dive into the structural machinery that lets you take a finite group apart and see why it has to look the way it does — Sylow theory and group actions are just the entry point, with most of the semester spent on nilpotent and solvable structure, coprime actions, and the local-to-global bridge that fusion and transfer provide. You work through this almost entirely via problem sets out of Isaacs and Kurzweil-Stellmacher, with one midterm and a final to make sure the proofs actually stick. It assumes you're comfortable with undergraduate algebra at the level of a first group theory course, and it's the standard prerequisite vocabulary for representation theory, cohomology of groups, and the local analysis that shows up in the classification of finite simple groups.

Credit3ECTS5FacultyFaculty of ScienceBölümMathematics

Değerlendirme 100% — 3 adım

30%
30%
40%
Midterm Midterm 30%
Final Final 30%
Homework Homework 40%

Önerilen kaynaklar 2 kitap

📖
Önerilen
Finite Group Theory
Martin Isaacs
2008 · American Mathematical Society
📖
Önerilen
The Theory of Finite Groups: an introduction
Hans Kurzweil, Bernd Stellmacher
2003 · Springer

Haftalık müfredat 14 hafta

Hafta 1
Introduction to group actions and p-groups: - Group actions, orbit-stabilizer theorem and Burnside's counting lemma. - Dihedral, semidihedral, generalized quaternion groups and some extra special p-groups
Hafta 2
Sylow Theorems and Nilpotent groups: - Proofs of the Sylow Theorems and the Frattini Argument. - Nilpotent groups and the Fitting Subgroup
Hafta 3
Some non-simplicity theorems: Brodkey's theorem and introduction to the Chermak-Delgado measure
Hafta 4
Subnormality: Wielandt’s zipper lemma and Baer’s theorem
Hafta 5
Group extensions and Schur-Zassenhaus Theorem: Extensions and semidirect products, Schur-Zassenhaus Theorem and its applications
Hafta 6
Solvable groups: Hall subgroups and Hall-Higman’ theorem, Carter’s subgroup
Hafta 7
Coprime actions: Glauberman’s lemma and the fixed points of coprime action
Hafta 8
Midterm and Review: Midterm exam review and preparation
Hafta 9
Commutators: Hall-Witt identity, three subgroups lemma and, extra special p-groups
Hafta 10
Transfer: Transfer homomorphisms and Transfer evaluation lemma
Hafta 11
Fusion: Burnside fusion theorem and weakly closed subgroups
Hafta 12
Normal p-complement theorems: Focal subgroup theorem, Frobenius and Burnside normal p-complement theorems
Hafta 13
Resistant p-groups: Some examples and proof that metacyclic p-groups are resistant when p is odd
Hafta 14
Frobenius groups: Fixed point free actions and construction of Frobenius groups

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⚠️ FZ engelleyen şartlar

Course Learning Outcomes: Course Learning Outcome Assessment Understand and apply key group-theoretic tools such as group actions, p-groups, Sylow theorems, and the Frattini argument. Midterm Homework Analyze solvable finite group structures using Hall subgroups, Carter subgroups and Schur-Zassenhaus theorem. Midterm Final Homework Investigate group structure by the fusion theory. Final Homework

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