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Course Outlines

PURE MTH 4107 - Groups and Rings - Honours

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.

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  • General Course Information
    Course Details
    Course Code PURE MTH 4107
    Course Groups and Rings - Honours
    Coordinating Unit 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
    Restrictions Honours students only
    Assessment Ongoing assessments, exam
    Course Staff

    Course Coordinator: Dr Stuart Johnson

    Course Timetable

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

  • 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)
    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
    Career and leadership readiness
    • 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
    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