MATHS 320 : Algebraic Structures


2022 Semester Two (1225) (15 POINTS)

Course Prescription

This is a framework for a unified treatment of many different mathematical structures. It concentrates on the fundamental notions of groups, rings and fields. The abstract descriptions are accompanied by numerous concrete examples. Applications abound: symmetries, geometry, coding theory, cryptography and many more. This course is recommended for those planning graduate study in pure mathematics.

Course Overview

This course is intended for students who have enjoyed the examples of algebraic structures presented in MATHS 254/255, 253 and/or MATHS 328, and who are interested in a unified treatment of many different mathematical structures. The main focus of the course is on the fundamental algebraic structures of groups, rings and fields. The abstract descriptions are accompanied by numerous concrete examples. Emphasis is placed on the contexts in which basic structures occur, methods by which they can be modelled and analysed, and their diverse applications. This course is recommended for those planning graduate study in pure mathematics. A student completing this course will have a basic knowledge of the fundamental of algebraic systems, and be able to construct simple proofs in an algebraic setting and appreciate the strength the strength of abstract algebraic methods.

Course Requirements

Prerequisite: MATHS 250, and MATHS 254 or 255

Capabilities Developed in this Course

Capability 1: Disciplinary Knowledge and Practice
Capability 2: Critical Thinking
Capability 3: Solution Seeking
Capability 4: Communication and Engagement
Capability 5: Independence and Integrity
Graduate Profile: Bachelor of Science

Learning Outcomes

By the end of this course, students will be able to:
  1. Display basic knowledge of fundamental algebraic systems: groups, rings and fields and their properties by solving relevant questions and examples (Capability 1 and 2)
  2. Display ability to analyse and answer questions, through examples and methods (Capability 1, 2 and 3)
  3. Demonstrate understanding of connections with other disciplines and real world applications by the use of relevant examples (Capability 1, 2, 3 and 4)
  4. Demonstrate understanding of the nature of proof and precision in mathematics, and the ability to present arguments both formally and informally in written form (Capability 2, 4 and 5)


Assessment Type Percentage Classification
Tutorials 10% Individual Coursework
Assignments 20% Individual Coursework
Test 25% Individual Test
Final Exam 45% Individual Examination
Assessment Type Learning Outcome Addressed
1 2 3 4
Final Exam

A minimum mark of 35% must be obtained in the final exam to pass the course

Key Topics

Week 1: Symmetric operations and groups
Week 2: Subgroups (including cyclic subgroups)
Week 3: Cyclic groups; Permutation groups;
Week 4: Group isomorphisms; Direct products; Inner products; Cosets;
Week 5: Lagrange’s Theorem; The Orbite-Stabiliser Theorem; Burnside's Theorem; Normal subgroups and factor groups
Week 6: Group homomorphisms; Finite abelian groups; The Cauchy theorem;
Week 7: Rings and integral domains
Week 8: Ideals, factor rings and ring homomorphisms
Week 9: Polynomial rings and factorisation
Week 10: Vector spaces (including extension fields)
Week 11: Field extensions (including algebraic extensions)
Week 12: Finite fields. 

Special Requirements

This course has no special requirements.

Workload Expectations

This course is a standard [15] point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect [36] hours of lectures, a [11] hour tutorial, [36] hours of reading and thinking about the content and [67] hours of work on assignments and/or test preparation.

Delivery Mode

Campus Experience

Attendance is [expected] at scheduled activities including [tutorials] to [receive credit for] components of the course. However it will be available for remote students with special circumstances (for example those outside New Zealand).
Lectures will be available as recordings. Other learning activities including tutorials will not be available as recordings.
Attendance on campus is [required] for the [test/exam].
The activities for the course are scheduled as a [standard weekly timetable].

Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The recommended textbook is "Contemporary Abstract Algebra", by J. A. Gallian (published by D.C. Heath & Company), 8th edition or later, but also detailed course notes will be provided, and these will include definitions, examples, properties and applications of the key concepts developed in the course.

Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

Academic Integrity

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

Class Representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.


The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website

Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page

This should be done as soon as possible and no later than seven days after the affected test or exam date.

Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter


Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 28/10/2021 08:01 p.m.