PURE MTH 7059 - Groups and Rings
North Terrace Campus - Semester 1 - 2022
General Course Information
Course Code PURE MTH 7059 Course Groups and Rings Coordinating Unit School of Mathematical Sciences Term Semester 1 Level Postgraduate Coursework Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Available for Study Abroad and Exchange Y Assumed Knowledge PURE MTH 2016 Course Description 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 Coordinator: Dr Stuart Johnson
The full timetable of all activities for this course can be accessed from Course Planner.
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.
University Graduate Attributes
This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:
University Graduate Attribute Course Learning Outcome(s)
Attribute 1: Deep discipline knowledge and intellectual breadth
Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.
Attribute 2: Creative and critical thinking, and problem solving
Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.
Attribute 3: Teamwork and communication skills
Graduates convey ideas and information effectively to a range of audiences for a variety of purposes and contribute in a positive and collaborative manner to achieving common goals.
Attribute 4: Professionalism and leadership readiness
Graduates engage in professional behaviour and have the potential to be entrepreneurial and take leadership roles in their chosen occupations or careers and communities.
Attribute 8: Self-awareness and emotional intelligence
Graduates are self-aware and reflective; they are flexible and resilient and have the capacity to accept and give constructive feedback; they act with integrity and take responsibility for their actions.
Recommended ResourcesJ. 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 LearningAssignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
Learning & Teaching Activities
Learning & Teaching ModesLecture 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 Workshops/Quizzes 33 82 Tutorials 12 24 Assignments 4 32 Group Project 1 18 Total 156
Learning Activities Summary
Lecture Schedule Week 1 Groups Definitions and Examples Week 2 Groups Cosets and Normal Subgroups Week 3 Groups New Groups from Old I: Factor Groups Week 4 Groups New Groups from Old II: Product Groups Week 5 Groups Finitely Generated Abelian groups Week 6 Groups Group Actions on Sets Week 7 Groups The Sylow theorems Week 8 Groups Rings, Fields and Integral Domains Week 9 Rings Polynomial rings Week 10 Rings Ideals and factor rings Week 11 Rings Eudlidean domains and Principal Ideal Domains Week 12 Rings Unique Factorisations Domains
The University's policy on Assessment for Coursework Programs is based on the following four principles:
- Assessment must encourage and reinforce learning.
- Assessment must enable robust and fair judgements about student performance.
- Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
- Assessment must maintain academic standards.
Assessment task Task type Due Weighting Learning outcomes Examination Summative Examination period 55% All Homework assignments Formative and summative Weeks 3,5,9,13 20% All Group Project Formative and summative Week 11 10% All Mid Semester Test Formative and summative Week 7 10% All Quizzes Formative Every week 5% All
Assessment Related RequirementsAn aggregate score of 50% is required to pass the course.
Assessment task Set Due Weighting Assignment 1 Week 1 Week 3 5% Assignment 2 Week 3 Week 5 5% Assignment 3 Week 7 Week 9 5% Assignment 4 Week 11 Week 13 5% Group Project Week 3 Week 11 10% Quizzes Weekly Weekly 5%
SubmissionAll 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) Grade Mark Description 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.
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