PURE MTH 3007 - Groups and Rings III
North Terrace Campus - Semester 1 - 2014
General Course Information
Course Code PURE MTH 3007 Course Groups and Rings III Coordinating Unit Pure Mathematics Term Semester 1 Level Undergraduate Location/s North Terrace Campus Units 3 Contact Up to 3 hours per week Prerequisites MATHS 1012 Assumed Knowledge PURE MTH 2106 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
2. Appreciate and be able to prove the basic results of group theory
and ring theory.
3. Understand and be able to apply the fundamental theorem of finite
4. Understand Sylow's theorems and be able to apply them to prove
elementary results about finite groups.
5. Appreciate the significance of unique factorization in rings and
6. Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty. 7. 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) Knowledge and understanding of the content and techniques of a chosen discipline at advanced levels that are internationally recognised. 1,2,3,4,5 The ability to locate, analyse, evaluate and synthesise information from a wide variety of sources in a planned and timely manner. 2,5,6 An ability to apply effective, creative and innovative solutions, both independently and cooperatively, to current and future problems. 3,4,6 Skills of a high order in interpersonal understanding, teamwork and communication. 7 A commitment to continuous learning and the capacity to maintain intellectual curiosity throughout life. 1,2,3,4,5,6,7 An awareness of ethical, social and cultural issues within a global context and their importance in the exercise of professional skills and responsibilities. 7
Recommended ResourcesJ. B. Fraleigh, “A first course in abstract algebra", covers most of the material in the course in a similar manner to that presented in lectures. 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 ModesThe lecturer guides the students through the course material in 30 lectures. Students are expected to engage with the material in the lectures. Interaction with the lecturer and discussion of any difficulties that arise during the lecture is encouraged. Students are expected to attend all lectures. In fortnightly tutorials students present their solutions to assigned exercises and discuss them with the lecturer and each other. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.
The information below is provided as a guide to assist students in engaging appropriately with the course requirements.
Activity Quantity Workload Hours Lectures 30 90 Assignments 5 30 Tutorials 6 36 Total 156
Learning Activities Summary
Lecture Schedule Week 1 Groups Groups and subgroups. Week 2 Groups Permutation groups, isomorphisms, cosets and normal subgroups. Week 3 Groups Conjugation, simple groups, homomorphisms and factor groups. Week 4 Groups The first isomorphism theorem, the Jordan-Hölder theorem. Week 5 Groups Products of groups, finitely generated Abelian groups. Week 6 Groups Groups acting on sets. Week 7 Groups The Sylow theorems and applications. Week 8 Rings Rings, subrings, integral domains and fields. Week 9 Rings Polynomials, ideals, factor rings. Week 10 Rings Factorisation in integral domains (Euclidean domains, principal ideal domains, unique factorisation domains). Week 11 Rings Theorems on integral domains and their proofs. Week 12 Rings Completion of proofs, revision.
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 70% All Homework assignments Formative and summative Weeks 3,5,7,9,11 20% All Tutorial participation Formative Weeks 2,4,6,8,10,12 10% All
Assessment Related RequirementsAn aggregate score of 50% is required to pass the course.
Assessment task Set Due Weighting Tutorial exercises 1 Week 1 Week 2 see below Assignment 1 Week 2 Week 3 4% Tutorial exercises 2 Week 3 Week 4 Assignment 2 Week 4 Week 5 4% Tutorial exercises 3 Week 5 Week 6 Assignment 3 Week 6 Week 7 4% Tutorial exercises 4 Week 7 Week 8 Assignment 4 Week 8 Week 9 4% Tutorial exercises 5 Week 9 Week 10 Assignment 5 Week 10 Week 11 4% Tutorial exercises 6 Week 11 Week 12
It is expected that each student will present twice in the tutorials. Each presentation will be worth 3%, the additional 4% is for attendance and participation in the other tutorials. This may have to be adjusted depending on enrolment.
SubmissionHomework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused 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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