Lectures: There will be 5 lectures and 1 practice class/tutorial each fortnight. ...
Peter J. Cameron, Introduction to Algebra, Oxford Univerity Press, Oxford 1998.
Course Information 3P5: Groups and Symmetry (MATH3335) Semester 1, 2009 School of Mathematics and Statistics The University of Western Australia Credit: 6 points The theme of this unit is the theory of groups and their use in measuring symmetry with special emphasis on geometric examples. The goal is for students • to understand the basic properties of groups and group actions; • to understand how theory is developed from basic axioms; • to understand mathematical proofs in elementary group theory; • to be able to prove basic facts about groups independently.
Unit Outcomes: Students are able to extend their knowledge of mathematical concepts and techniques and adapt known solutions to different situations; understand and appreciate the power and beauty of mathematical abstraction; communicate effectively with others; present mathematical results in a logical and coherent fashion; and undertake continuous learning, aware that an understanding of fundamentals is necessary for effective application.
Unit coordinator/Lecturer: Michael Giudici Contact details:
[email protected], Room 211 Maths building, x3351. Please email or visit me if you have any questions about the course. I am usually in my office but if you can’t find me please email to arrange a time.
Lectures: There will be 5 lectures and 1 practice class/tutorial each fortnight. Tuesday 3pm and 4pm Mathematics Lecture Room 2 Thursday 11am Mathematics Lecture Room 2
Texts and recommended reading: The main text is Joseph A. Gallian, Contemporary Abstract Algebra, Houghton Mifflin. 1
The bookshop should have the 7th edition. Earlier editions such as the 5th and 6th are perfectly acceptable. The 5th has been placed on closed reserve in the library. Warning: Some of my notation will differ from the book, in particular how I compose functions and multiply permutations. The following books have been placed on 3-day loan in the library and are good substitutes for Gallian but also have the same problem with functions and permutations. • John B. Fraleigh, A first course in abstract algebra, 7th edition, Addison-Wesley, Boston. (The library also has earlier editions which are suitable.) • M. A. Armstrong, Groups and symmetry, Springer-Verlag New York 1988. There are many other books in the library which cover the material of the course. They generally have call numbers 512.02 or 512.2. You should be careful as to what notation they use. Two books in the library which do use my notation for composition of functions and permutations are: • Peter J. Cameron, Introduction to Algebra, Oxford Univerity Press, Oxford 1998. • Abraham P. Hillman and Gerald L. Alexanderson, Abstract algebra:a first undergraduate course, 5th edition PWS, Boston 1994.
Unit webpage: All assignments and handouts will be posted at http://www.maths.uwa.edu.au/Units/math3335-s1-2009-crawley/
Handbook Entry: http://units.handbooks.uwa.edu.au/units/math/math3335
Assessment: 70% exam, 20% tests, 10% assignments. There will be two tests (each worth 10%) and two assignments (each worth 5%). If you miss a test then you will get a mark of zero (0) unless you can provide a medical certificate. Dates and times for the tests and assignments will be announced on the unit webpage. Late assignments which are at most 2 days late will lose 50% and after that will receive a mark of zero (0).
Faculty Policies: Please ensure you are familiar with the Faculty’s policies: Calculators: http://www.ecm.uwa.edu.au/studentnet/exams/calculators Plagiarism: http://www.ecm.uwa.edu.au/studentnet/exams/dishonesty Appeals: http://ecm.uwa.edu.au/studentnet/exams Scaling: http://www.ecm.uwa.edu.au/staffnet/policies/assessment Supplementary Exams: http://ecm.uwa.edu.au/studentnet/exams/supps Charter of Student Rights and Responsibilities: http://www.secretariat.uwa.edu.au/home/policies/charter
Plagiarism: Plagiarism or any other form of cheating is treated severely. The School belongs to the Faculty of Engineering and Mathematical Sciences, and is bound 2
by the Faculty’s policy on plagiarism, see Web link above. Note that the penalties described in it apply whether you are enrolled in this faculty or not.
Learning requirements: Any first unit in abstract mathematics is hard, as the abstract axiomatic approach may seem very unfamiliar at first. It is therefore very important that you work through the material in each lecture again by yourself. Aim to understand every proof and be able to reproduce it. It is essential that you attempt all exercise questions by yourself. It is also very helpful to study in groups and explain methods and concepts to each other. However, make sure you do not just understand a solution to an exercise, but are also able to explain this solution to someone else. You know you understand the material if you can explain it. Hence to pass the unit it is recommended that you • attend all lectures • spend at least 1 hour revising a lecture afterwards - aiming for understanding • attempt all exercises by yourself • work through exercise solutions with other students in the unit • read the course related material
Course Outline: Basic definitions and properties, dihedral groups, integers modulo n. Subgroups, Cyclic groups. Permutation groups: cycle notation, even and odd permutations. Cosets, normal subgroups, Lagrange’s Theorem. Permutation groups: orbits, stabilisers, Orbit-Stabiliser Theorem. Homomorphisms, isomorphisms, automorphisms, conjugacy. Direct Products. Quotient groups, First Isomorphism Theorem. Fundamental Theorem of Finite Abelian Groups. Group actions, Cayley’s Theorem, Orbit-Counting Lemma. Sylow’s Theorems. Finite groups of rotations in R3 .
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