## PURE MTH 3007 - Groups and Rings III

### North Terrace Campus - Semester 1 - 2016

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 Groups and Rings III School of Mathematical Sciences Semester 1 Undergraduate North Terrace Campus 3 Up to 3 hours per week Y MATHS 1012 PURE MTH 2106 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 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.

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
• 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
None.
##### Recommended Resources
J. 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 Learning
Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni.
• Learning & Teaching Activities
##### 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.

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
 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.
• 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
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
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

Marks are allocated for tutorial attendance and participation, which will include a requirement for students to give presentations during tutorials, the precise details will depend on enrolment numbers and so will be determined at the beginning of the semester, at this time students will be given details of what will be required.
##### 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.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
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
• Fraud Awareness

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 studentâ€™s disciplinary procedures.

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