## PURE MTH 7059 - Groups and Rings

### North Terrace Campus - Semester 1 - 2021

The algebraic notions of groups and rings are of great interest in their own right, but knowledge and understanding of them is of benefit well beyond the realms of pure algebra. Areas of application include, for example, advanced number theory; cryptography; coding theory; differential, finite and algebraic geometry; algebraic topology; representation theory and harmonic analysis including Fourier series. The theory also has many practical applications including, for example, to the structure of molecules, crystallography and elementary particle physics. Topics covered are: (1) Groups, subgroups, cosets and normal subgroups, homomorphisms and factor groups, products of groups, finitely generated abelian groups, groups acting on sets and the Sylow theorems. (2) Rings, integral domains and fields, polynomials, ideals, factorization in integral domains and unique factorization domains.

• General Course Information
##### Course Details
Course Code PURE MTH 7059 Groups and Rings School of Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Up to 3 hours per week Y PURE MTH 2016 The algebraic notions of groups and rings are of great interest in their own right, but knowledge and understanding of them is of benefit well beyond the realms of pure algebra. Areas of application include, for example, advanced number theory; cryptography; coding theory; differential, finite and algebraic geometry; algebraic topology; representation theory and harmonic analysis including Fourier series. The theory also has many practical applications including, for example, to the structure of molecules, crystallography and elementary particle physics. Topics covered are: (1) Groups, subgroups, cosets and normal subgroups, homomorphisms and factor groups, products of groups, finitely generated abelian groups, groups acting on sets and the Sylow theorems. (2) Rings, integral domains and fields, polynomials, ideals, factorization in integral domains and unique factorization domains.
##### Course Staff

Course Coordinator: Dr Stuart Johnson

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
 1. Demonstrate understanding of the idea of a group, a ring and an integral domain, and be aware of examples of these structures in mathematics. 2. Appreciate and be able to prove the basic results of group theory and ring theory. 3. Understand and be able to apply more advanced results on groups: the fundamental theorem of finitely generated abelian groups, Burnside's theorem and the Sylow theorems. 4. Appreciate the significance of unique factorization in rings and integral domains. 5. Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty. 6. Demonstrate skills in communicating mathematics orally and in writing.

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)
Deep discipline knowledge
• informed and infused by cutting edge research, scaffolded throughout their program of studies
• acquired from personal interaction with research active educators, from year 1
• accredited or validated against national or international standards (for relevant programs)
1,2,3,4,5,6
Critical thinking and problem solving
• steeped in research methods and rigor
• based on empirical evidence and the scientific approach to knowledge development
• demonstrated through appropriate and relevant assessment
all
Teamwork and communication skills
• developed from, with, and via the SGDE
• honed through assessment and practice throughout the program of studies
• encouraged and valued in all aspects of learning
7
• technology savvy
• professional and, where relevant, fully accredited
• forward thinking and well informed
• tested and validated by work based experiences
7
Self-awareness and emotional intelligence
• a capacity for self-reflection and a willingness to engage in self-appraisal
• open to objective and constructive feedback from supervisors and peers
• able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
7
• Learning Resources
None.
##### Recommended Resources
J. B. Fraleigh, “A first course in abstract algebra", Addison-Wesley, 7th edition, 2002; covers most of the material in the course in a similar manner to that presented in lectures.

M. A. Armstrong, "Groups and Symmetry", Springer, 1988; covers most of the material about groups in the course, but in addition has many geometric applications and examples.

There are many other introductory texts on abstract algebra in the library which students may find useful as references.
##### Online Learning
Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
• Learning & Teaching Activities
##### Learning & Teaching Modes
Lecture notes and videos will be made available through MyUni. Students
are expected to work through these each week, following up with quizzes
and tutorials to reinforce the material and provided practical
experience at working with it. The lecturer will be available to help with weekly consulting sessions, and through interaction in the course discussion board.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload Hours Lectures/Quizzes 33 82 Tutorials 12 24 Assignments 5 30 Group Project 1 20 Total 156
##### Learning Activities Summary
 Week 1 Groups Groups and subgroups, generators and cyclic groups Week 2 Groups Permutation groups, homomorphisms and isomorphisms Week 3 Groups Cosets and normal subgroups, conjugation, simple groups Week 4 Groups Factor groups and the first isomorphism theorem, composition series Week 5 Groups Direct products of groups, decomposability, torsion group Week 6 Groups Finitely Generated Abelian groups Week 7 Groups Groups actions and Burnsides Theorem Week 8 Groups The Sylow theorems, Cauchy's Theorem and applications Week 9 Rings Introduction to rings, integral domains, polynomial rings Week 10 Rings Ideals and isomorphism theorems, maxial and principal ideals Week 11 Rings Eudlidean domains, Gaussian integers, PIDs and UFDs Week 12 Rings Polynomials over UFDs
In weeks 2,4,6,8,10 and 12 there will be a tutorial in the Wednesday class.
There will be a mid semester test, most likely in the Wednesday class in week 7 after the mid semester break. There will be one meeting during one of the classes for discussions about the group project.
##### Small Group Discovery Experience
A group project with a written report develops research skills, teamwork skills, and communication skills.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary
Examination Summative Examination period 50% All
Homework assignments Formative and summative Weeks 3,5,7,9,11 25% All
Group Project Formative and summative Week 12 10% All
Tutorial Participation Formative Every Week 5% All
Quizzes Formative Every week 10% All
##### Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
##### Assessment Detail
Assignment 1 Week 1 Week 3 5%
Assignment 2 Week 3 Week 5 5%
Assignment 3 Week 5 Week 7 5%
Assignment 4 Week 7 Week 9 5%
Assignment 5 Week 9 Week 11 5%
Tutorial Participation Weekly Weekly 5%
Group Project Week 3 Week 12 10%
Quizzes Weekly Weekly 10%

##### Submission
All work will be submitted electronically through MyUni.
Students may be elegible for an extension or exemption from an
assignment for medical or compassionate reasons. Documentation is
required and the lecturer must be notified as soon as possible.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
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