# MATH30400 Further Groups and Rings

In this module we introduce and develop some of the more advanced ideas in Group Theory and Ring Theory. A sample of topics covered are: Group actions, orbit-stabilizer theorem, Burnside’s lemma. Sylow theorems, other examples and applications. Linear groups. The Chinese Remainder Theorem. Matrices over EDs and PIDs, the Smith normal form. Modules, free modules and torsion modules. Finitely generated modules over a PID, and the structure theorem. Applications including the classification of finite abelian groups, Jordan canonical form, rational canonical form.

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Curricular information is subject to change

Learning Outcomes:

On completion of this module, the student is expected to:- demonstrate an understanding of the theoretical aspects covered in this module;- be able to carry out any computations involving the objects described in this module (such as finding the orbits of a group action, finding the Smith normal form of a matrix, decomposing a module into invariant factors, reducing a matrix to Jordan canonical form, for example).

Indicative Module Content:

Review of the first, second and third isomorphism theorems for groups and rings. Subgroups generated by a subset, generators and relations. Group actions, orbit-stabilizer theorem, Burnside’s lemma. Sylow theorems, other examples and applications. Linear groups.

Ideals and subrings generated by subsets, review of principal ideal domains, unique factorization domains, Euclidean domains. The Chinese Remainder Theorem. Matrices over EDs and PIDs, the Smith normal form. Modules, free modules and torsion modules. Finitely generated modules over a PID, and the structure theorem. Applications including the classification of finite abelian groups, Jordan canonical form, rational canonical form.

Student Effort Hours:
Student Effort Type Hours
Specified Learning Activities

36

Autonomous Student Learning

34

Lectures

24

Tutorial

6

Total

100

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Co-requisite:

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Examination: Midterm Exam 1 Week 6 Yes Standard conversion grade scale 40% No

20

No
Examination: Take-home exam (6-hours-long), to be scheduled on Week 13 Coursework (End of Trimester) Yes Standard conversion grade scale 40% No

60

No
Examination: Midterm Exam 2 Week 10 Yes Standard conversion grade scale 40% No

15

No
Assignment: Each assignment submission will receive 0.7 marks towards final grade. Throughout the Trimester n/a Standard conversion grade scale 40% No

5

No

Carry forward of passed components
No

Resit In Terminal Exam
Autumn Yes - 2 Hour
Feedback Strategy/Strategies

• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

A First Course in Abstract Algebra, by John Fraleigh (ISBN-10: 0201763907, ISBN-13: 978-0201763904)

Abstract Algebra, by David S. Dummit and Richard M. Foote (ISBN-10: 0471433349, ISBN-13: 978-0471433347)

Undergraduate Commutative Algebra: 29 (London Mathematical Society Student Texts, Series Number 29), by Miles Reed (ISBN-10: 0521458897, ISBN-13: 978-0521458894)
Name Role
Firtina Kucuk Tutor
Mr Peter Neamti Tutor