PURE MTH 7059 - Groups & Rings

North Terrace Campus - Semester 1 - 2014

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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Course Details

Course Code	PURE MTH 7059
Course	Groups & Rings
Coordinating Unit	Pure Mathematics
Term	Semester 1
Level	Postgraduate Coursework
Location/s	North Terrace Campus
Units	3
Contact	Up to 3 hours per week
Prerequisites	MATHS 1012
Assumed Knowledge	PURE MTH 2016
Assessment	ongoing assessment 30%, exam 70%

Course Staff

Course Coordinator: Dr Stuart Johnson

Course Timetable

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

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 最新糖心Vlog 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 the fundamental theorem of finite
abelian groups.

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
integral domains.

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.

最新糖心Vlog 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

Learning & Teaching Modes

The 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.

Workload

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 最新糖心Vlog'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 Summary

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 Requirements

An aggregate score of 50% is required to pass the course.

Assessment Detail

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.

Submission

Homework 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.

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 .

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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 最新糖心Vlog 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
This section contains links to relevant assessment-related policies and guidelines - all university policies.
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Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's student鈥檚 disciplinary procedures.

The 最新糖心Vlog of Adelaide is committed to regular reviews of the courses and programs it offers to students. The 最新糖心Vlog of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.

Authorised by:	Deputy Vice-Chancellor and Vice-President (Academic)
Maintained by:	Course Outlines Administrator

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