This Course Guide has been taken from the most recent presentation of the course. It would be useful for reference purposes but please note that there may be updates for the following presentation.
MATH S214
Fundamentals of Group Theory and Geometry
| Introduction |
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Welcome to MATH S214! This is a two-semester 10-credit course covering fundamental knowledge in group theory and geometry.
You will meet a lot of 'groups' in this course. Different groups have different elements but they share common properties. These properties will be generalised to form abstract properties of groups. With these abstract properties, theorems on groups have been developed. Group Theory studies these theorems. In MATH S214, you will be guided towards the fundamental properties of groups and explored them in details.
You will learn group action in the last unit of Group Theory. Through group actions, you will see geometry as a space with different matrix groups acting on it. You will study four types of geometry: affine, inversive, non-Euclidean and projective. Non-Euclidean geometry has its challenge to mathematicians and astronomers in the nineteenth century. Projective geometry has origins for artists to paint three-dimensional scenes onto two-dimensional pictures. You will learn the important theorems in each of this geometry and apply them in the course.
| Structure of the Course |
|
You will see M203 Introduction to Pure Mathematics on the front cover of the study units. This is because MATH S214 is taken from UK Open University's course M203.
MATH S214 has 3 study blocks, namely, the Introduction block, the Group Theory block and the Geometry block.
The Introduction block has 3 units: Units 2, 3 and 4. Unit 1 of M203 Introduction block is not included in this course. This unit is on Curve Sketching. It is irrelevant to the study of Group Theory and Geometry.
Unit 2 is on the language used in the course. It teaches you the notation and terms occurred in the study units. You must study this unit and refer your notation to it when necessary; however, this unit will not be assessed in assignments and examination.
Unit 3 starts with symmetries of a geometric object and states the axioms of groups. Unit 4 introduces you to subgroups of a group and to isomorphism between groups. You will learn how to relate the symmetry groups in Unit 3 to the modular arithmetic groups that you may have learned in other courses. You must be familiar with Abelian groups, cyclic groups and their properties after studying the Introduction block.
| Introduction block |
Title of the Unit |
| Unit 2 |
Mathematical Language
|
| Unit 3 |
Symmetry
|
| Unit 4 |
Groups and Subgroups
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The Group Theory block has 5 units. You start with cycle forms in Unit 1 and apply them to groups you learned in the Introduction block. You will be introduced to symmetric groups and alternating groups then.
You will learn Lagrange's theorem, normal subgroups and quotient groups in Unit 2. Unit 3 teaches you a special technique to justify whether a set is a normal subgroup or not. You will study how homomorphisms, quotient groups and isomorphisms are related in Unit 4. Unit 5 introduces you to group action concept.
| Group Theory block |
Title of the Unit |
| Unit 1 |
Permutations
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| Unit 2 |
Cosets and Lagrange's Theorem
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| Unit 3 |
Conjugacy |
| Unit 4 |
Isomorphisms and Homomorphisms
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| Unit 5 |
Group Actions
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Finally, the Geometry block discusses four types of geometry. The German mathematician Felix Klein proposed in 1872 that Geometry is a space with different transformations to map different figures onto others. When you study a special type of geometry, you must make sure what the space is and what the group of transformation is under study. Remember the important theorems in each geometry and compare them among different geometries.
| Geometry block |
Title of the Unit |
| Unit 1 |
Affine geometry
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| Unit 2 |
Inversive geometry
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| Unit 3 |
Non-Euclidean geometry
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| Unit 4 |
Projective geometry: lines
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| Unit 5 |
Projective geometry: conics
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| Notes on the Course |
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Structure of the Units
The Introduction to each unit gives a brief outline of the various sections of that unit and indicates how the sections depend on one another. Most units have four or five sections, each corresponding very roughly to three evenings work (some longer, some shorter!). You should be able to complete the study of each unit within three weeks. The Presentation Schedule, will help you plan your study so that by following it you will be able to submit your assignments by the cut-off dates and complete the course with time to spare for revision.
Group Theory Model
You will find 4 yellow sheets in the course material. They are named Group Theory Model. You are expected to use scissors and sellotape to build five Platonic models from these four sheets. The models will help you to visualise symmetries in 3-dimensions that are taught in Unit 3 of the Introduction block.
Television Programmes
MATH S214 has 13 television programmes. A television programme is indicated by a TV symbol in the study units. This means that you should watch the TV programme for the unit while working on the section. The TV section contains the text directly associated with the programme; if the television programme introduces materials not covered elsewhere in the unit, these materials will be briefly given in the text, to help you re-study the section later. There may be some preparatory materials to read, or problems to attempt, before watching the programme, so you should look at the television section before actually watching the programme. These sections may also contain materials to study after watching the programme, and occasionally extra materials not directly linked to the programme.
The television programmes usually last for 25 minutes. You will need to plan your study schedule so that you can watch the relevant programme at the appropriate time. Although the TV notes will give you some idea about the programme, they are not meant to replace it. You are strongly advised to watch these programmes if you are to get full benefit of the materials. MATH S214 is centred on abstract concepts that will not always be easy to grasp. The TV programmes try to help by showing you concrete examples and guiding you through the more demanding parts of the units.
All MATH S214 television programmes are broadcast on the TVB Pearl Channel on Sunday mornings and are also available at the OUHK Library.
Audio programmes
MATH S214 has 7 audio CDs. They contain audio programmes that is characterised by 'frames' in the study units.
You must listen to these audio lessons. You will frequently need to stop the listening to solve a problem or answer a question. The lesson has indicated these points by a few bars of music. Make sure that the 'pause' or 'stop' button on your machine is within easy reach while you are working on an audio section.
Problems and exercises
There are many problems for you to solve as you work through the texts. The solutions to the problems are collected together in an appendix at the end of each unit. The final section of each unit consists of a collection of exercises whose solutions are also given in an Appendix. The problems and exercises are provided to test your understanding of the material and practise the various techniques you have learned in the unit.
Numbering
The numbering of Theorems, Problems, and so on, starts at 1 at the beginning of each section. For example, for a complete reference we would refer to Problem 3 of Section 1 (within the same unit), to Problem 3 of Unit 2, Section 1 (within the same Block), or to Problem 3 of Group Theory block, Unit 2, Section 1 (in a different Block).
Course Handbook
The Course Handbook provides a convenient source of the definitions, results, theorems and strategies in the units. Youwill not be allowedto bring the Course Handbook to the exam. Another copy of the handbook will be given to you together with the exam paper.
Calculators and mathematical software
You can use a calculator, or mathematical software such as Scientific Notebook, to evaluate mathematical functions such as exponential, logarithmic, trigonometric and hyperbolic when you study the course.
Calculators are allowed in the examination. The University has a List of Approved Models of Calculators so that students realise which types of calculators are allowed in the examination. This list is updated based on the types of calculators approved by the Hong Kong Examinations and Assessment Authority. You will receive the list from the Registry before the examination.
Presentation Schedule
The Presentation Schedule gives full details of the week in which you are advised to study the unit. It shows which units are covered by which TMAs, the cut-off dates for the assignments, the titles of the television and audio programmes associated with each unit, and the weeks which have been left free for you to catch up with your work. It also gives the provisional dates for tutorials. Unless told otherwise you may assume these dates are correct.
Computer Access
This course requires you to log on the Online Learning Environment through the Internet regularly (see Section 5.2). The PC Labs in the OUHK main campus and the Island Learning Centre have these facilities available.
| Assessment |
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MATH S214 has 4 tutor-marked assignments (TMAs) for continuous assessment and a three-hour final examination. The continuous assessment and the examination count equally towards your final grade for the course. All TMAs are required, that is, your marks on these assignments are all counted. If you do not submit an assignment, a zero mark for that assignment will be assumed in your record. To summarise:
- each TMA contributes 25% towards your overall continuous assessment score (OCAS);
- the continuous assessment contributes 50% towards your course score;
- the examination contributes the other 50% towards your course score.
To obtain the 10 credits from the course, you must obtain at least 40 marks in the OCAS and at least 40 marks in the examination.
Consider a student having the following marks in his/her continuous assessment record.
| |
TMA01 |
TMA02 |
TMA03 |
TMA04 |
| scores
|
56 |
65 |
48 |
0
|
This student scored 0 mark on TMA04. It is not clear whether the student got 0 on this assignment or the student did not submit this assignment.
The OCAS for MATH S214 is the average of the four assignment scores. The OCAS of this student is:
OCAS = (56+65+48+0) / 4 = 42.25 marks
The student met the requirement for continuous assessment. If the student obtained at least 40 marks in the examination, he/she would be awarded a pass for the course.
Continuous assessment
Format of the assessment
The TMAs are related to the study units as follows:
| TMA01 |
Units 3 and 4 of Introduction block |
| TMA02 |
Units 1, 2 and 3 of Group Theory block |
| TMA03 |
Units 4 and 5 of Group Theory block;
Units 1 and 2 of Geometry block |
| TMA04 |
Units 3, 4 and 5 of Geometry block |
Your TMAs
The number of marks assigned to each part of a question is printed in the right-hand margin of the question, and this should give you a rough indication of the relative difficulty or importance of each part of a question.
Tutors are asked to comment fully on each question and to use every opportunity to teach on the basis of your solutions. Comments by your tutor are regarded as a major part of the teaching of the course. Use the TMAs for revision purposes before the examination.
If you have difficulty with one or more questions on a TMA or if you have difficulty in meeting the cut-off date, seek help and advice from your tutor.
Doing assignments is important for learning new materials. The University encourages students to discuss assignment questions with one another. However, you have to submit your own working for assessment purposes. Students are not allowed to release or share their solutions, right or wrong. If two assignment scripts are considered to be essentially the same, their scores will be changed to zero if the students cannot give a reasonable explanation.
Final examination
Preparing for the examination
Make sure that you schedule your revision of the course materials over as long a period as possible, so that you will not run short of time as the examination day approaches. You will find the Course Handbook and the Exercises section in each unit particularly useful in your revision.
Get into the habit of using your Course Handbook as you will be allowed to take it into the examination.
Specimen examination paper
A specimen examination paper will be provided before the examination, for which sample solutions will also be provided. The specimen gives you a general idea on the format of the examination paper and the level of the examination questions.
| Support during the Course |
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Keeping up to schedule
Try to keep up to the Presentation Schedule. There may be times when you fall behind and you need to make decisions on how to catch up. If you fall far behind schedule for any reasons, contact your tutor or the course coordinator for advice.
Adult learners sometimes do not have enough time to complete all questions in a TMA. If you have finished at least three quarters of it before the cut-off date, you are advised to submit what you have done, rather than to ask for a late permission. Do not spend too much time on just one TMA. You are expected to study the whole course.
The Online Learning Environment (OLE)
Through the OLE, you can get the current course information and communicate with other students, tutors and the course coordinator. The OLE User Guide explains the hardware and software requirements for you to access the OLE. It also helps you use the components in the OLE.
Stop Press Notices
From time to time you will receive Stop Press Notices giving you up-to-date information on any changes in the running of the course. These notices are placed in the OLE. You must read these notices carefully and remember to change your schedules accordingly.
Comments and Errata
We welcome your comments on the course materials, and would be grateful to be told of any errors or misprints that you notice.
When you receive an Errata, in print or online, you should correct the error(s) in the relevant course materials as early as possible. Otherwise, you may waste time on unnecessary calculations or understanding.
Tutorials
Details for tutorials such as venue and time are given in the Tutorial Schedule.
You should consider tutorials as opportunities to discuss problems with your tutor and course-mates. Although the tutor might spend some time to review the major concepts presented in the units, he/she will not lecture. This is not his/her role and he/she will not have time to do so. Tutorials are not substitutes for lectures. All the lecturing is conducted through the study units, the television programmes and the audio programmes. It is your responsibility to let your tutor know which part of the learning topics you would like him/her to explain or to demonstrate more at tutorials.
Come to tutorials, discuss with your tutor and get along with other students on the course. Jot down your questions when you study and let your tutor know the questions in advance. Your tutor will be there to give you advice in tutorials.
Support from your tutor
Your tutor helps you at tutorials and gives you comments on your TMAs. You can also reach your tutor through the telephone or the OLE.
You are encouraged to seek help from your tutor; however, you need to be considerate. A tutor is usually allocated 35 students and he/she will have to divide his/her time among them. Moreover, he/she is likely to be in full time employment in addition to his/her commitment to the University. This will restrict his/her availability. You will be told at what times your tutor can be contacted through the telephone and you should aim to keep to those times.
Whenever possible you should try to keep telephone conversations from becoming too long. Remember that you may be preventing another student from contacting the tutor. Sometimes it may be more sensible to write down your problems and discuss them at a tutorial.
The interactive tools in the OLE provide you another opportunity to get advice from your tutor. Do participate in the OLE Discussion Board. Your tutor and course-mates will share their views on the learning topics there.
Course coordinator
If you have any queries that your tutor cannot settle, he/she may advise you to contact the course coordinator. The course coordinator is a staff member of the School of Science and Technology in the University. Details about the course coordinator can be found in the OLE.
| Start here |
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Before you start working on the study units, you are given a week to get familiar with the OLE. The last page of the OLE User Guide states where you can get technical support. If you have been familiar with the OLE, then you should start on Unit 2 of the Introduction block without necessarily waiting for another week as suggested in the Presentation Schedule. An earlier start gives you more flexibility in studying the course.
