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MATH 429

Finite Group Theory

MATH 429 takes the basic group theory you saw in abstract algebra and pushes it into the structural machinery working mathematicians actually use: Sylow analysis, nilpotent and solvable decompositions, coprime actions, and the transfer-and-fusion toolkit that controls how p-subgroups sit inside a finite group. Work is theorem-driven — you'll spend the semester proving things on homeworks and two exams, gradually building from group actions up to results like Schur-Zassenhaus, Burnside's fusion theorem, and the classification of Frobenius groups. It's a natural follow-up to a first algebra course and the standard on-ramp for anyone heading toward representation theory, cohomology of groups, or research in algebra.

Credit3ECTS5FacultyFaculty of ScienceBölümMathematicsPreMATH 323

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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