PURE MTH 3007 - Groups and Rings III

North Terrace Campus - Semester 1 - 2022

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 3007
    Course Groups and Rings III
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Undergraduate
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3 hours per week
    Available for Study Abroad and Exchange Y
    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 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.
    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.

    1,2,3,4,5,6

    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.

    all

    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.

    7

    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.

    7

    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.

    7
  • Learning Resources
    Required 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.

    Workload

    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
  • 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
    Assessment taskTask typeDueWeightingLearning outcomes
    Examination Summative Examination period 55% All
    Homework assignments Formative and summative Weeks 3,7,13 15% All
    Group Project Formative and summative Week 11 10% All
    Mid Semester Test2 Formative and summative Weeks 5, 9 15% All
    Quizzes Formative Every week 5% All
    Assessment Related Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    Assessment taskSetDueWeighting
    Assignment 1 Week 1 Week 3 5%
    Assignment 2 Week 5 Week 7 5%
    Assignment 3 Week 11 Week 13 5%
    Test 1 Week 5 Week 5 7.5%
    Test 2 Week 9 Week 9 7.5%
    Group Project Week 3 Week 11 10%
    Quizzes Weekly Weekly 5%


    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.

    Course Grading

    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.

  • 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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