# MST30010 Group Theory and Applications

This is an intermediate level group theory module, with an emphasis on arithmetic applications. Topics covered include:

1. Groups, examples of groups (with an emphasis on cyclic groups)
2. Products of groups
3. Subgroups, group homomorphisms, quotient groups (abelian case)
4. Chinese Remainder Theorem formulated in terms of product groups, arithmetic applications
5. Group-theoretic proof of Euler's theorem and Chinese Remainder Theorem revisited for the multiplicative groups
6. Group structure of multiplicative groups modulo N, quadratic residues, Gauss' Reciprocity Law
7. Solving polynomial equations modulo primes (e.g. Chevalley--Warning Theorem)
8. Discrete logarithm problem and data security

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Curricular information is subject to change

Learning Outcomes:

Upon successful completion of this module a student should have the following skills:

1. Solid understanding of the main concepts and results indicated within the contents of this module.
2. The ability to apply counting arguments involving Lagrange's theorem and the class equation to deduce statements on the structure of groups.
3. The ability to compare group (via homomorphisms between them).
4. The ability to use the isomorphism theorems to identify groups.
5. The ability to build combinatorial and arithmetic arguments relying on fundamentals of group theory.

Indicative Module Content:

Basics of groups, subgroups, cyclic groups, cosets, Lagrange's theorem, normal subgroups and quotient groups, the centre of a group, homomorphisms, the isomorphism theorems, the subgroup correspondence theorem, group actions, orbits, Cayley's theorem, inner automorphisms, conjugacy, the class equation, Cauchy's theorem and p-groups.

Student Effort Hours:
Student Effort Type Hours
Specified Learning Activities

32

Autonomous Student Learning

40

Lectures

18

Tutorial

12

Total

102

Approaches to Teaching and Learning:
Lectures
Enquiry & Problem-based Learning
Requirements, Exclusions and Recommendations
Learning Requirements:

Students should have completed a course in abstract algebra prior to taking this module (for example, MST20010). This course should contain at least an introduction to groups, subgroups, cosets, and permutation groups.

All questions about eligibility (in particular if you think the meet the requirements but have not passed MST20010) should be addressed to the module coordinator.

Module Requisites and Incompatibles
Co-requisite:
MST20010 - Algebraic Structures

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Examination: Final Exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

60

Assignment: 4 homework assignments Varies over the Trimester n/a Standard conversion grade scale 40% No

20

Examination: Midterm Exam (on Week 8, tentative) Unspecified No Standard conversion grade scale 40% No

20

Carry forward of passed components
No

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

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment
• Self-assessment activities

How will my Feedback be Delivered?

Self-assessment activities are tentative and will be based on the following model: Students will be asked to grade their own responses to select assignments, based on a detailed solution sheet. They will prepare a detailed report on their weaknesses on key aspects of the module. Based on the accuracy of this report, their homework grades will be positively updated. The other two feedback strategies to employed (Feedback individually to students, post-assessment and Group/class feedback, post-assessment) are self-explanatory.

Title: A first course in abstract algebra
Author: John B Fraleigh
Name Role
Dr Daniele Casazza Tutor
Konstantinos Maronikolakis Tutor