Undergraduate Abstract Algebra
- If geometry lets us see what we are thinking about, algebra enables us to talk precisely about what we
see, and above all to calculate. Moreover it tends to organize our calculations and to conceptualize them; this in turn can lead to further geometrical construction and algebraic calculation. -M.W. Hirsch
- Abstract Algebra: An Undergraduate Course Rings and Fields Robert Howlett University of Sydney (PG) Exactly what the title says it is, nothing more, nothing less-albeit somewhat unorthodox in structure. Howlett also intends the notes to be used as an introduction to rigorous mathematics and there's appropriate emphasis on naive set theory, functions and logic in the early sections. What will in general be a bit unusual to modern American mathematics students is the emphasis on rings rather then groups as the prototypical algebraic structure. Algebracists know that the emphasis on rings was common in both the early history and textbooks on the subject. It makes logical sense as rings are the generalization of the integers and makes algebra a natural outgrowth of number theory. Still, many-including the author of this site-think pedagogically, this approach is problematic.The greater complexity of the structure of rings may make it rougher going for the beginner then one that begins with groups and builds forward. Be that as it may, the notes are generally well written and well problemed with good examples.
- Two other unusual aspects of the notes that make them unique are : a) The emphasis on field extensions as a general means of building new rings and fields. You almost never see this in undergraduate sources and Howlett does it quite well here. b) The very early introduction of ruler and compass constructions are a motivator of general field theory- this is done even before the author has defined rings! Matrices and number systems are used throughout as the examples, which is very smart for a first course. All in all, a very well done and original presentation of sophisticated material to beginners in algebra. Highly recommended!
- Abstract Algebra Lecture Notes Steven Tschantz Vanderbilt University Spring 2001 (PG)A quite careful and somewhat stylistically verbose introduction to abstract algebra with standard coverage. He focuses on groups, rings and fields with the majority of emphasis on groups with the other structures being built on top of groups. As I said, I think pedagogically this is the best way to proceed. Tschanz fleshes out the notions of both careful proof and algebraic structures very well if pedantically, he gives many examples and insights along the way. That being said, he makes a rather strange choice at the beginning of the notes-he chooses to introduce the notion of "algebra" in the sense of universal algebra, of which the usual structures of groups, rings, fields, etc. are all special cases. I'm not sure if this is a good idea to fiost into beginners, especially since the term algebra has a quite different meaning in general algebra i.e. a vector space with a defined multiplication mapping in addition to the binary operations of vector addition and scalar multiplication. I think the term algebraic structure would be more appropriate and less confusing. Overall, though, he writes very clearly and motivates the constructions and theorems of algebra quite well for the beginner. There are lots of good problems, ranging from simple computations to thinking problems. Some surprises are present- for example, he finishes with some simple applications of field theory to coding. Best of all, at 80 pages, this is definitely a source that can be covered either in class or by self study in one semester and doesn't try to do too much. Highly recommended as an introduction.
- Abstract Algebra Benedict Gross Harvard University.transcribed by George Vasmer Leverett 2010 (PG-13) I was pretty excited to see that transcriptions of Gross' OpenCourseWare lectures on algebra at Harvard were available online. I've never personally watched Gross lecture and I have to make it my buisness to watch the online recorded lectures one day. Friends of mine who have been privileged enough to be in his classes have had nothing but praise for him as a teacher. In any event, this was clearly not a standard abstract algebra class for typical undergraduates-it's clearly an intensive course for much stronger students. Contents: Week 1.Review of linear algebra.Groups Examples of groups Basic properties and constructions Week 2 Permutations. Cosets, Z/nZ.Week 3.Quotient groups, first isomorphism Theorem.Abstract fields, abstract vector spaces.Construction and invariants of vector spaces.Week 4.Abstract linear operators and how to calculate with them.Properties and construction of operators.Week 5 Orthogonal groups.Week 6.Isometries of plane figures.Cyclic and dihedral groups.Finite and discrete subgroups of symmetry groups.Week 7.Group actions.Basic properties and constructions.Groups acting on themselves by left multiplication.Groups acting on themselves by conjugation.Week 8.A5 and the symmetries of an icosahedron.Sylow theorems.Study of permutation groups.Week 9.Rings.Examples of rings.Basic properties and constructions.Week 10.Quotient rings, extensions of rings.Integral domains, fields of fractions.Week 11 Special lecture.Week 12.Euclidean domains, PIDs, UFDs.Gauss’lemma.Eisenstein’s criterion.Algebraic integers.Week 13.Structure of ring of integers in a quadratic field.Dedekind domains.Ideal class groups.Week 14.Wrap-up.(incomplete,finish)
- Abstract Algebra Bruce Ikenaga University of Millersville 2014 (PG) These are the course notes and other materials for a first undergraduate algebra course focusing on groups and rings. But to say that completely understates the wonderful cache of materials that the author has written and set up at the home page for the course to the point of misrepresenting it. Every student of algebra wishes his or her instructor had put the care and effort into preparing course materials that Ikenaga has placed here for his students. First and foremost, Inkenaga has written a beautiful, very careful and complete set of lecture notes for the course. They are rigorous and concise without being terse-limiting his coverage to groups and rings only really allows him be both detailed and relatively brief simultaneously. He also gives many standard examples from geometry,number theory and analysis in complete detail. For example, he gives fully detailed developments of both the Euclidean symmetry groups and the polynomial rings Indeed, when given a choice between a detailed theorem proof and a detailed example, he often opts to spend his time developing the example. I'm completely down with this-giving students a big stock of well-understood examples is really a teacher's function and how those students really come to understand a new mathematical subject. He also stocks the site with many good exercises and detailed review sheets, which many serious students will appreciate, particularly those engaging in self-study. There are many very instructive pictures. If all instructors of abstract algebra put as much time,passion and effort into preparing course materials, then students would have no reason to fear the course other then their own laziness. An outstanding resource for both students and teachers of algebra and very highly recommended.
- Abstract Algebra Paul Melvin Bryn Mawr College Fall 2011 (PG-13) Clear but extremely dry and condensed set of lecture notes to supplement an algebra course based on Dummit and Foote's Abstract Algebra.I've never been a huge fan of this book to begin with, as I've stated at my blog, because it's always seemed like a dessicated, bloated, diluted version of Herstien's classic Topics in Algebra.These accompanying notes are even more dry and tasteless, shooting along at a breakneck speed in bullet point form with virtually no expository chit chat.That being said, despite the brevity, they give a very complete outline of the book with all the main points stated clearly and effectively. I wouldn't use this as a course text, but it might prove quite useful for review come test time. For those looking for a study or review guide to a first course in abstract algebra, this just might be what the doctor ordered.
- Abstract Algebra D. S. Malik Creighton University John N. Mordeson Creighton University M.K. Sen Calcutta University (PG) A surprisingly comprehensive, well written and careful set of notes for a substantial undergraduate course in abstract algebra. Contents: Sets, Relations, and Integers Introduction to Groups Permutation Groups Subgroups and Normal Subgroups Homomorphisms and Isomorphisms of Groups Direct Product of Groups Introduction to Rings Subrings, Ideals, and Homomorphisms Ring Embeddings Direct Sum of Rings Polynomial Rings Euclidean Domains Unique Factorization Domains Maximal, Prime, and Primary Ideals Modules and Vector Spaces Field Extensions Multiplicity of Roots Finite Fields The deliberately limited choice of topics is extremely classical and chooses depth over breadth in the presentation. This results in a text which is not only quite flexible, but extremely readable and surprisingly sophisticated for the undergraduate level.There are many detailed concrete examples and accompanying pictures, such as for the symmetry groups of the square in the Euclidean plane.2 aspects of the presentation are particularly worth noting: While there are many examples presented in great depth, there are also step by step solved problems at the end of each section, called Worked Out Examples. These are quite useful for the beginner, especially since the notes are presented at a level of sophistication greater then one normally sees. The other good touch are the several well-written and researched historical notes and commentaries about the origins of certain concepts of algebra and the mathematicians that discovered them inserted throughout. I've always believed science is a human story which is best understood in the context of the questions and/or events that motivated the specific aspects of it's creation.All in all, this is an excellent text for a serious abstract algebra course for undergraduates and it's highly recommended to all.
- Algebraic Structures I Fall 2003 Emil Volcheck Loyola Marymount University (PG) A very nice, comprehensive set of "bullet point" notes for the first semester of a year long abstract algebra course.Contents: Naive set theory, permutations, matrices, basic number theory, groups subgroups, homomorphisms and isomorphisms, cosets, Lagrange's theorem, quotient structures, Abelian groups and direct products, cyclic groups, group actions and the Orbit Stabilizer Theorem and symmetry groups.It focuses almost entirely on naive set theory, group theory and number theory. Indeed, much of the material is very similar in style-although pitched at a somewhat lower level-to the wonderful second chapter of Herstien's classic text Topics in Algebra, from which I first learned it. There are many concrete examples and very good exercises from the website independent of the course text which was Joseph Gallian's very popular text-although there are virtually no pictures. The author takes a more purist approach to algebra, which is fine if one is careful to supply good examples and exercises for the beginner, as Volcheck does. The one drawback of the course materials linked here is that they cover only group theory-no other algebraic structures are covered. This may make them of limited use in a rapidly paced first course or a course that does not begin with group theory. (Originally, abstract algebra courses at the undergraduate level began with rings, following the example of Saunders MacLane and Garrett Birkoff, who created the first modern algebra course for undergraduates at Harvard University in 1941. Some courses still do this.) In any event,the course materials created by Volcheck and which he has so graciously allowed to remain available online over the years will make a wonderful study supplement for either the group theory part of an American algebra sequence or an undergraduate group theory course in the United Kingdom. Highly recommended-within its limits.
- Elementary Abstract Algebra by W.Edwin Clark University of South Florida 2001 (PG) An excellent, focused and brief online text for a one semester abstract algebra course. Contents Binary Operations Introduction to Groups The Symmetric Groups Subgroups The Group of Units of Zn Direct Products of Groups Isomorphism of Groups Cosets and Lagrange s Theorem Introduction to Ring Theory Axiomatic Treatment of R N Z Q and C The Quaternions The Circle Group A Some Rules of Logic B Functions C Elementary Number Theory D Partitions and Equivalence Relations The book clocks in at 105 pages-a more then reasonable length for a text that can be covered from beginning to end in a one semester course. As I said in the introduction, the other main problem with most standard academic textbooks, other then price, is they're too damn long to effectively cover in one or two semesters. A book like Clark's is far more realistic in length and coverage and demonstrates clearly one of the main advantages of online textbooks-without publisher pressure to make the book as long as possible, people tend to write books that cover just what they need and nothing more. This book covers basic set theory, groups and rings-and that's it. Period. You'd bet even with such limited topics, such a book would be extremely dense and with almost no examples. And you'd lose that bet-there's a ton of important if standard examples with each new concept. The secret is the extremely narrow selection of topics-that's what allows Clark to be so brief without being terse and sparse in detail. Indeed, this book may have the opposite problem-it may be too brief for some teachers! I think most of us would agree,though-it would be very difficult to cover more then this in one semester. Not only is the book very focused and reasonably brief, it's extremely well organized. Definitions and theorems are stated and proved extremely carefully and clearly, but concisely. It also seems to me it would be a perfect introduction for non-mathematics majors, such as math ed teachers or computer science students. I was surprised in the course of finding this that more people weren't aware of it and it'd be in widespread use, even "under the table", as an undergraduate text. But no-this manuscript seems nearly unheard of. Well, until now, of course.Highly recommended for self study or as a one semester course text.
- Abstract Algebra done Concretely Donu Arapura Purdue University February 19, 2004 (PG-13) These are Donu Arapura's lecture notes for a one year undergraduate abstract algebra course.The title refers to the emphasis that Arapura wanted to put on the role that each of the usual algebraic structures plays not only in algebra itself, but other areas of mathematics. Looking at these quite intriguing notes, I can see why none of the standard books appealed to Arapura. His selection of topics and their development is deep, sophisticated and above all, decidedly nonstandard for an undergraduate course in algebra. Contents 1 Natural Numbers 2 Principles of Counting 3 Integers and Abelian groups 4 Divisibility 5 Congruences 6 Linear Diophantine equations 7 Subgroups of Abelian groups 8 Commutative Rings 9 A little Boolean Algebra* 10 Fields 11 Polynomials over a Field 12 Quotients of Abelian groups 13 Orders of Abelian groups 14 Linear Algebra over Zp* 15 Nonabelian groups 16 Groups of Permutations 17 Symmetries of Platonic Solids 18 Counting Problems involving Symmetry* 19 Proofs of theorems about group actions 20 Groups of 2 × 2 Matrices 21 Homomorphisms between groups 22 Groups of order 1 through 8 23 The Braid Group* 24 The Chinese remainder theorem 25 Quotients of polynomial rings 26 The finite Fourier transform* 27 Matrix Representations of Groups* 28 The ring of Quaternions* 29 Quaternions and the Rotation Group* A Sets and Functions B Maple What initially struck me about the notes was how Arapura continually points out several subtle but critical mathematical facts that usually aren't pointed out to beginners-which a lot of mathematics students really wish their professors would.For example, right off the bat, he shows how to prove the principle of mathematical induction from the Well Ordering principle of the natural numbers. Another very unusual aspect of the notes is the great importance that is placed on the relationship between basic algebra and number theory. Not only are most of the examples drawn from number theory, most of the applications are,too. Abelian groups are introduced as a direct generalization of the integers under addition. Which brings us to another important original quality of the notes-Arapura uses the title as a statement of method throughout-that is to say, he always introduces all concepts in the least general and most concrete manner possible while still maintaining high rigor.Abelian groups are used throughout most of the notes instead of general abstract groups. This leads to the introduction of commutative rings rather then general rings,which leads naturally to a discussion of fields and so forth. The very gradual "loosening" of conditions leading to slow generalization of concepts is not only helpful for students learning the material, it mirrors to considerable degree the actual historical development of the subject.There are also several important but sophisticated topics that are usually not taught at this level included in the optional starred sections, such as braid groups and matrix transformations.Arapura also uses MAPLE code to implement the course concepts quite a bit, both in the examples and the exercises.The notes, overall, are quite engaging and lucid, with many examples and a large number of good exercises.The highly unorthodox nature of this algebra course is a great demonstration of what's possible when a professor takes teaching a course to motivated students seriously and decides to develop his or her own program for the course from scratch. Granted,this is sadly something most professors are neither encouraged or rewarded for doing,especially those at high pressure research-oriented departments. But a text like this should be exhibit A when we're trying to make the case why they should be encouraged to do so. Students of algebra will love learning from these notes and teachers looking for a course to challenge strong students with may find just what they're looking for here. Most highly recommended!
- Abstract Algebra I Galois Theory David Wilkins Trinity College 2013 (PG-13)
- Abstract Algebra II Module Theory, Commutative Rings and Algebraic Number Theory David Wilkins Trinity College 2013 (PG-13) These are the course notes for the revised year long course in algebra at Trinity. Earlier, shorter versions of the course can be found here. I strongly recommend students with no abstract algebra background check out the 1996 version of the notes. These notes really assume a good working knowledge of linear algebra and some knowledge of the elements of groups and rings. Contents:(MA3411 Semester 1) Basic Principles of Group Theory 2 Basic Principles of Ring Theory 3 Polynomial Rings 4 Field Extensions 5 Ruler and Compass Constructions 6 Splitting Fields and the Galois Correspondence 7 Roots of Polynomials of Low Degree 8 Some Results from Group Theory 9 Galois's Theorem concerning the Solvability of Polynomial Equations (MA3412 Semester 2) Section 1: Commutative Rings and Polynomials Section 2: Integral Domains Section 3: Noetherian Rings and Modules Section 4: Determinants and Integral Closures Section 5: Discrete Valuations over Dedekind Domains Section 6: Finitely-Generated Modules over Principal Ideal Domains It's hard to imagine these notes being useful for a general undergraduate abstract algebra course in the United States, except at the very best schools. They don't cover enough or at a sufficiently sophisticated level for a first year graduate course either. So how these would be used in an actual course outside Europe would be somewhat problematic. They are well written and very careful, but extremely concise and somewhat dry. They remind me in coverage and style of I.M. Herstein's classic Topics in Algebra in a lot of ways- except they are much drier and not as pleasant to read.They also focus much more on field and non-commutative ring theory then Herstien's classic.The discussions of ruler and compass constructions and module theory over commutative rings in particular are very nice. But the big defect in these notes are really what differentiates them from Herstien above all and not in a positive manner- they have very few
examples and even fewer exercises.There are some,but nowhere near as many as a course of this level of difficulty would require.Despite being well written, between the awkward level and the near absence of exercises, I'd be hard pressed to recommend them to a student for self study, even a very strong one. I might recommend them as supplementary reading- but that's about it.
- Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University August 16, 2013 (PG) This is a complete revision and updating of a textbook for a standard abstract algebra course which Judson originally published in 1994. He has very generously made it available as an Open Source Sage Textbook as part of the Open Textbook Initiative project. It's based on various courses in algebra Judson has been teaching at Austin State for many years. Not only has Judson extensively proofread and corrected the original, he has included a large number of important applications that have arisen in the field, particularly in computer science,since the original edition was published. These applications have been very significant in both the author's teaching and research.The result is a remarkably strong and versatile textbook that can be used effectively for many different kinds of courses in abstract algebra at the undergraduate level. Preface 1 Preliminaries 2 The Integers 3 Groups 4 Cyclic Groups 5 Permutation Groups 6 Cosets and Lagrange's Theorem 7 Introduction to Cryptography 8 Algebraic Coding Theory 9 Isomorphisms 10 Normal Subgroups and Factor Groups 11 Homomorphisms 12 Matrix Groups and Symmetry 13 The Structure of Groups 14 Group Actions 15 The Sylow Theorems 16 Rings 17 Polynomials 18 Integral Domains 19 Lattices and Boolean Algebras 20 Vector Spaces 21 Fields 22 Finite Fields 23 Galois Theory Hints and Solutions GNU Free Documentation License Notation Index The book assumes very little in the way of actual prerequisites-the student basically has to know how to manipulate 2 x 2 matrices and that's about it.It's considerably more comprehensive then can be covered in even a 2 semester course-again, the general problem of "brain diarrhea"I talked about in the introduction.But in this case, it's helpful because it's clear from the introduction and how the book is structured that the author doesn't intend for people to use the whole book-just the parts they want. This is what gives the book it's enormous versatility. Many standard topics are done exceptionally well-the sections on permutation groups, cryptography and the fields are particularly strong. There are many clear examples in each section. Each section has a very large collection of exercises that rang in difficulty from simple computations to substantial proofs. There are optional advanced topics for honors courses, such as lattices and the Sylow theorems.Also,Judson is very careful keep the book at an introductory level while still maintaining a very high level of rigor.As icing on the cake, each chapter comes with historical commentary and references for further study. Judson is to be applauded for writing an amazingly complete and beautifully written introduction to abstract algebra at the undergraduate level that's both freely available and available in cheap paperback edition.It's a textbook that will serve any undergraduate course in algebra superbly for either self study or as a very reasonably priced course text. You really have nothing to complain about with this book. The highest possible recommendation for either teachers or students of algebra at any level!
- Honors Abstract Algebra Course Notes Math 55a Curtis McMullen Harvard University Fall 2009 (PG-13) This are course notes for the algebra half of the infamous Math 55 at Harvard referred to above.It's a rigorous and mercilessly paced introduction to linear and abstract algebra for gifted freshman.Contents 1 Introduction 2 Set Theory 3 Vector spaces 4 Polynomials 5 Linear Operators 6 Inner product spaces 7 Bilinear forms 8 Trace and determinant 9 Introduction to Group Theory 10 Symmetry 11 Finite group theory 12 Representation theory 13 Group presentations 14 Knots and the fundamental group The coverage of the notes is, as one would expect,amazingly broad. Basically, this is the bare minimum amount of algebra a mathematics major freshman in principle would have to learn in order to begin taking graduate level courses the following year. McMullen-a very skilled teacher and lecture note writer-does his best to make this bed of nails as comfortable as possible for his students, but there's really no way to do that when a course is this intense. Still, the notes are well written and manage to cram a ridiculous amount of algebra into 77 pages. (He and his TA's may add more detail and diagrams in the actual lectures, but if they do, there's no record of them here.) You'd think such notes would be relentlessly dry, but McMullen does a pretty good job peppering the notes with brief but motivating remarks that help clarify critical points. Linear algebra is mostly presented in terms of abstract operators There are some applications-for example, nilpotent graphs and there's a brief treatment of the fundamental group of knots.Unfortunately-and in this case, it's a big problem for those who dare to use these notes for self study-there are no exercises. All the exercises are assigned out of the 4 references for the course-I'll leave you to look them up at the homepage link. This plus the light-speed coverage of these notes pretty much makes them useless by themselves for self-study. If you're a strong student looking for a challenge in self study or a professor about to tackle an honors course-you'll have to supplement these notes extensively to even begin to use them. I'd recommend either Gross or Arapura's notes instead-or Ash's wonderful Dover paperback, supplemented with Stoll or one of the other strong linear algebra course notes above. These would make for a much more human manageable course.Download and study them if you dare, but you've been warned.
- Algebra I lecture notes written by Anton Stefanek Imperial College from the Group Theory And Linear Algebra lectures of Martin Liebeck at Cambridge(PG-13)
- Algebra 2 lecture notes written by Anton Stefanek Imperial College from the Group Theory And Linear Algebra lectures of Martin Liebeck at Cambridge (PG-13) This is a very beautiful and comprehensive set of lecture notes that are unusual right off the bat in the accreditation of sources. Stefanek, a graduate student in computer science, was a joint mathematics and computer science major as an undergraduate at Cambridge and proceeded to type up all the lecture notes in mathematics he took then. He's done an outstanding job expanding and developing his own version of the notes using the "official" notes as the skeleton he later fleshed out. Since these abstract algebra notes are derived from courses at the second year University of Cambridge Tripos, you'd expect them to be more sophisticated then the average undergraduate abstract algebra course in the US. And you'd be right. But the real joy here is how
- Stefanek takes the usual compressed lecture notes of Cambridge, which vary widely in their level of detail, and "unravels" them into wonderfully detailed presentations containing lucid proofs and loads of examples-and does it in such a way that the original conciseness of the notes isn't lost. Just enough detail is added to bring the ordinarily dry notes to mathematical life-it makes one feel like one is reading an actual real time transcript of Liebeck's lectures. Remarkably clear,detailed and lucid while remaining parsimonious with words.While the choice of material is rather standard, the organization is not. The first term discusses the more concrete aspects of group theory and linear algebra, such as group tables, the symmetric groups, matrix algebra and arithmetic and basic vector space theory.The second semester deals with more sophisticated topics that rely more on the relationship between abstract vector space theory and group theory, such as isomorphisms,Cauchy's Theorem with applications to combinatorics, a careful treatment of determinants and much more. The big drawback of these otherwise wonderful notes are-you guessed it-no exercises. Sadly,I suspect most of the original exercises where solved by Stefanek and their solutions incorporated into the text as the many examples in the typed version. With this price now paid for the depth and readability of this version, the users of these notes will need to find exercises elsewhere. I'm certain that won't be too difficult- indeed, I'm certain many universities in the UK run some version of this course and it won't be too difficult to find supplementary exercises there. Stefanek has done all algebra students a huge favor by writing up and making these notes available to algebra students everywhere. Very highly recommended! Contents (Algebra 1 )Groups 1.1 Definition and examples 1.1.1 Group table 1.2 Subgroups 1.2.1 Criterion for subgroups 1.3 Cyclic subgroups 1.3.1 Order of an element 1.4 More on the symetric groups Sn 1.4.1 Order of permutation 1.5 Lagrange's Theorem 1.5.1 Consequences of Lagrange's Theorem 1.6 Applications to number theory 1.6.1 Groups 1.7 Applications of the group Zp 1.7.1 Mersenne Primes 1.7.2 How to find Meresenne Primes 1.8 Proof of Lagrange’s Theorem 2 Vector Spaces and Linear Algebra 2.1 Definition of a vector space 2.2 Subspaces 2.3 Solution spaces 2.4 Linear Combinations 2.5 Span 2.6 Spanning sets 2.7 Linear dependence and independence 2.8 Bases 2.9 Dimension 2.10 Further Deductions 3 More on Subspaces 3.1 Sums and Intersections 3.2 The rank of a matrix 3.2.1 How to find row-rank(A) 3.2.2 How to find column-rank(A)? 4 Linear Transformations 4.1 Basic properties 4.2 Constructing linear transformations 4.3 Kernel and Image 4.4 Composition of linear transformations 4.5 The matrix of a linear transformation 4.6 Eigenvalues and eigenvectors 4.6.1 How to find evals / evecs of T? 4.7 Diagonalisation 4.8 Change of basis 5 Error-correcting codes 5.1 Introduction 5.2 Theory of Codes 5.2.1 Error Correction 5.3 Linear Codes 5.3.1 Minimum distance of linear code 5.4 The Check Matrix 5.5 Decoding (Algebra 2) I Groups 1 Introduction 2 Symmetry groups 2.1 Symmetry groups 2.2 More on D8 3 Isomorphism 3.1 Cyclic groups 4 Even and odd permutations 4.1 Alternating groups 5 Direct Products 6 Groups of small size 6.1 Remarks on larger sizes 6.2 Quaternian group Q8 7 Homomorphisms, normal subgroups and factor groups 7.1 Kernels 7.2 Normal subgroups 7.3 Factor groups 8 Symmetry groups in 3 dimensions 9 Counting using groups II Linear Algebra 10 Some revision 11 Determinants 11.1 Basic properties 11.2 Expansions of determinants 11.3 Major properties of determinants 11.3.1 Elementary matrices 12 Matrices and linear transformations 12.0.2 Consequences of 12.1 12.1 Change of basis 12.2 Determinant of a linear transformation 13 Characteristic polynomials 13.1 Diagonalisation 13.2 Algebraic & geometric multiplicities 14 The Cayley-Hamilton theorem 14.1 Proof of Cayley-Hamilton 15 Inner product Spaces 15.1 Geometry 15.2 Orthogonality 15.3 Gram-Schmidt 15.4 Direct Sums 15.5 Orthogonal matrices 15.6 Diagonalisation of Symmetric Matrices
- A Semester Course in Basic Abstract Algebra Marcel B. Finan Arkansas Tech University December 29, 2011(PG)Another excellent, careful if very standard online textbook by Finan for a first undergraduate course in abstract algebra.If you're looking for something different and original, look elsewhere, you won't find it here. But what you will find is a very solid first course on all the standard algebraic structures and their major theorems-all done very carefully and with lots of examples and exercises. And let's face it-isn't that really all we need from a good textbook?And you can't beat the price with a baseball bat.It's a typical Finan online text-crystal clear, rigorous without being abstract,lots of illuminating examples and tons of good problems that aren't too difficult. You could do a lot worst then pick this one as a self-study text or a cheap book for your students if teaching abstract algebra. Highly recommended. Contents 0 Preliminary Notions 1 The Concept of a Mapping 2 Composition. Invertible Mappings 3 Binary Operations 4 Composition of Mappings as a Binary Operation 5 Definition and Examples of Groups 6 Permutation Groups 7 Subgroups 7.1 Definition and Examples of Subgroups 7.2 The Alternating Group 8 Symmetry Groups 9 Equivalence Relations 10 The Division Algorithm. Congruence Modulo n 10.1 Divisibility. The Division Algorithm 10.2 Congruence Modulo n 11 Arithmetic Modulo n 12 Greatest Common Divisors. The Euclidean Algorithm 13 Least Common Multiple. The Fundamental Theorem of Arithmetic 14 Elementary Properties of Groups 15 Generated Groups. Direct Product 15.1 Finitely and Infinitely Generated Groups 5.2 Direct Product of Groups 16 Cosets 17 Lagrange's Theorem 18 Group Isomorphisms 19 More Properties of Isomorphisms 20 Cayley's Theorem 21 Homomorphisms and Normal Subgroups 22 Quotient Groups 23 Isomorphism Theorems 24 Rings: Definition and Basic Results 25 Integral Domains. Subrings 26 Ideals and Quotient Rings
- ALGEBRA ABSTRACT AND CONCRETE EDITION 2.6 FREDERICK M. GOODMAN 2014. (PG-13) Yet another free online textbook for undergraduate linear or abstract algebra. Not that I'm complaining, mind you-the more, the better! But it's important to keep in mind that while the situation will hopefully change in the future, most online free "textbooks" in mathematics are unpolished lecture notes with all their imperfections. So the availability of Goodman's book is a very good thing indeed. It's instructive to compare Goodman's book with some of the other lecture notes and online books we've seen so far at this level: Unlike Arapura's notes, which essentially presents algebra as the direct generalization of number theory, Goodman's book focuses much more on the relationship between abstract algebra and classical Euclidean geometry. In fact, like Arapura and more comprehensively, Judson's outstanding online book, Goodman deftly weaves together theory and applications.It's quite similar in scope and topic selection to Judson.But ironically, despite the deliberately intuitive emphasis of Goodman's book, it's style and aims are quite different from Judson. There are far fewer applications and the book goes considerably further into advanced topics in algebra, such as multilinear algebra, module theory and Galois theory then Judson's book. Also,unlike Judson, it does presume students have a good grasp of classical Euclidean geometry and basic linear algebra. The author uses both constantly and without review throughout. Goodman also uses commutative diagrams fluently..While I don't think it has sufficient sophistication and range to use for a first year graduate course, it certainly goes quite a bit deeper then the average undergraduate text. In some ways, it's intended for a more ambitious course then Judson. As a result, for the average self-study student, Judson might be a better choice of the two texts. It would depend on where the interests of the student or teacher lie.In any event, the book is beautifully written with many examples and good exercises. I particularly like his presentations of group actions and modules and how he interweaves them. I was first introduced to the conception of a vector space as that of a "field of scalars acting on the abelian group of vectors" by Kenneth Kramer in my honors algebra course some years ago-I'm very happy to see Goodman use it for both vector spaces and R-modules. Overall, it's quite similar in spirit to Micheal Artin's Algebra although it's level of presentation is somewhat lower. Indeed, it covers more topics then Artin's classic! Goodman has written a wonderful and versatile book here that can be used for many different kinds of undergraduate courses in algebra, from a one semester course for high school teachers to a full blown one year course in algebra for honors students. An outstanding text all around and the highest possible recommendation. Contents Preface The Price of this Book A Note to the Reader Chapter 1. Algebraic Themes Chapter 2. Basic Theory of Groups Chapter 3. Products of Groups Chapter 4 Symmetries of Polyhedra Chapter 5 Actions of Groups Chapter 6 Rings Chapter 7 Field Extensions – First Look Chapter 8 Modules Chapter 9 Field Extensions – Second Look Chapter 10 Solvability Chapter 11. Isometry Groups Appendix A. Almost Enough about Logic Appendix B Almost Enough about Sets B.1. Families of Sets; Unions and Intersections B.2. Finite and Infinite Sets Appendix C. Induction C.1. Proof by Induction C.2. Definitions by Induction C.3. Multiple Induction Appendix D Complex Numbers Appendix E Review of Linear Algebra E.1 Linear algebra in Kn E.2. Bases and Dimension E.3. Inner Product and Orthonormal Bases Appendix F. Models of Regular Polyhedra Appendix G. Suggestions for Further Study Index
- Abstract Algebra II Nathan Dunfield Math 418 University of Wisconsin Urbana Champlain Spring 2010 Course Materials (PG-13) A pretty standard set of handwritten notes for the second semester of a year-long undergraduate algebra course. Dunfield writes well and the notes are clear and readable and he supplies many good exercises. The major asset here is a good introduction to the motivation behind classical algebraic geometry in the study of cubics and conics in the plane. But to be honest, there's nothing here that can't be found done even better in Miles Ried's classic Undergraduate Algebraic Geometry or even better for free, William Fulton's online classic text. Feel free to download the notes and peruse them- but I don't think there's anything here you can't get done better in some of the other sources.
- Algebra: RINGS, FIELDS AND GALOIS THEORY ANDREW TULLOCH University of Sydney (R) I had to double check when I saw these notes that I was reading it right. But I was. These are notes for an advanced undergraduate/first year graduate course in abstract algebra-20 pages total and completely omitting proofs. How's that for terse notes for an active learning course?It's tempting to dismiss them, but consider this: These notes are well organized, the definitions are clear and they have quite a few good examples. They're quite abstract-they begin with monoids and take it from there. For a first year graduate student preparing for their prelims in algebra, I think working through these notes and seeing how many of the details her or she can fill in would be an excellent preparation. But other then mathematical geniuses, I can't see any students that can actually learn algebra from scratch from these notes. Recommended for review and intensive studying only.
- Honors Algebra I-II: Abstract Algebra Fall 2006/Spring 2007 Ulrich Meierfrankenfeld Michigan State University September 1, 2009 (PG-13) These are Meierfrankenfeld's lectures for a 2 semester honors course in algebra. The structure of the notes is fairly classical-you won't find R-modules,homological algebra, categories or tensor products anywhere in here. Content-wise, he's going strictly by the old Van der Waerden mantra-groups, rings and fields. That being said, the style of the notes is quite modern-there are commutative diagrams and many of the examples focus on the role of these structures in geometry. A significant role is also played by group actions. I found them quite careful, lucid, abstract and challenging. In fact, Meierfrankenfeld is sometimes bafflingly careful. For example, at the beginning of the notes, he gives Zermelo's construction of the natural numbers from the axioms of set theory. It's a great presentation, better then what I see in most set theory texts.But what it has to do with abstract algebra is still a mystery to me.The notes are exceedingly rich in examples, with many careful pictures. In many ways, the notes reminded me-again!-of I.M. Herstien's Topics in Algebra Unfortunately, as is the case with many of the best course notes still online from the original course, the author hasn't made the original homework sets available along with the notes. That greatly undermines these notes as a stand-alone text. That being said, I think it might be worth the effort for an instructor to assemble problem sets and to use it as a course text. And for self study students, the exercises in the aforementioned Herstien would be a perfect accompaniment. Be warned-they are definitely not for the average student taking abstract algebra. But for strong undergraduates, they'll make an excellent course text or study source. Very highly recommended! Contents 1 Preface 1.1 Sets 1.2 Functions and Relations 2 Groups 2.1 Definition and Examples Elementary Properties of Groups 2.3 Subgroups 2.4 Lagrange's Theorem 2.5 Normal subgroups 2.6 Homomorphisms and the Isomorphism Theorems 2.7 Group Actions 2.8 Sylow p-subgroup 3 Rings 3.1 Definitions and Examples 3.2 Elementary Properties of Rings 3.3 Homomorphism and Ideals 3.4 Polynomials Rings 3.5 Euclidean Rings 3.6 Primes in Integral Domains 3.7 The Gaussian integers 3.8 Constructing Fields from rings 4 Field Theory 4.1 Vector Spaces 4.2 Field Extensions 4.3 Splitting Fields 4.4 Separable Extension 4.5 Galois Theory 4.6 Fundamental theorem of Algebra 4.7 Geometric Construction 4.8 Wedderburn's Theorem A.1 The Binomial Theorem
- Introduction to Algebra Alexander Fink Queens College of the University of London 2013 (PG) Standard UK style first course in abstract algebra. Covers all the basics concisely but not tersely. Interestingly, there are 2 versions of the lecture notes for the course available: A scanned PDF version of Fink's actual handwritten notes taken down in real time by one of the students and a TeX-ed version of the official 2012 notes by Konstantin Ardakov The real strength of these notes-both versions-are the exercises. There are many creative exercises that range from simple computations to head scratchers-and none too difficult-that build on the material in the notes and develop topics not directly covered-such as the general solution of the quintic closed algebraic equation, the geometry of the quaternions, proving the modular integer subgroup Z91 is not a field and many more. These exercises will prove very valuable to self study students as well as for coursework. As for the notes themselves-both cover a good standard course in algebra, but the handwritten notes by Fink are legible and cover considerably more. But without the exercises-well, it's a pretty vanilla course in algebra. In other words, a good solid introduction to abstract algebra without all the bells and whistles.The bells and whistles are in the problem sets and tests and it's strongly recommended students do them. Recommended especially for the exercises. Contents: Proofs and logic. Irrationality of 2. Division algorithm. 1. Induction 2. Complex numbers: Basic algebraic operations (fields). Argand diagram. Euler's notation. Extracting roots from complex numbers. Statement of Fundamental Theorem of Algebra. 3. Other number systems: Pseudo-complex numbers, hyper-complex numbers, quaternions. 4. Sets and Relations. Finite directed graphs. Partitions. Equivalence relations. Relationship between surjections and equivalence relations. 5. Congruence modulo m. Modular arithmetic.6. Operations. Rings. Skew fields. Matrices. 7. Groups.
- Groups of units. Subgroups, cyclic groups, Lagrange's Theorem. Permutations, symmetric group, sign.
- Elements of Abstract and Linear Algebra by Edwin H. Connell (PG) This free Open Source online text is fascinating-not only for the actual text contents, but the overriding motivations and thoughts of the author on writing it. This was an attempt by Connell to craft an algebra book that was a) actually readable from cover to cover in at most 2 semesters,b) emphasized linear algebra and didn't separate it from abstract algebra, c) encourages active learning on the part of the students while not leaving them to develop the entire course themselves and d) was as cheap as humanly possible for the students. I'm happy to report all four of the author's goals have been met-although I do think there's room for improvement. As one might expect in a book this short that covers a relatively large amount of material, many of the less difficult results are left for the student to prove. That's ok because Connell shows very good judgement what statements are too difficult for beginners to prove and what can be safely left for them to prove. He also does some very innovative things in the presentation that are well worth considering. (Which is rather ironic since the author says in the introduction that one of his goals was to "avoid all innovation." Seems he achieved some in spite of himsef.) For example, he begins to develop group theory on the integers under addition without ever defining a group. He develops basic number theory in terms of subgroups of Z by defining a subgroup as a nonempty subset of Z closed under addition and inverses. This allows him to not only develop basic number theory in the context of subgroups of Z, it's then very straightforward to develop abstract groups as generalizations of this material. Also, he gives a very careful development of basic set theory and defines associativity, the inverse, and identity in terms of mappings. The book appears to be heavily influenced-AGAIN,especially in the group chapter-by I.M. Herstien's Topics in Algebra -right down to the writing of functions on the right for permutations.It is interesting and important to note that while many of the chapters are independent of each other, the one indispensable chapter is the chapter on groups. This is because Connell builds all other algebraic structures and their basic results as extensions of groups. It's hard to argue that this is the most straightforward and natural way to build a first course on algebra. And this pretty much sums up what makes this book work so well-it's a straight walk through abstract algebra with no sidebars or digressions via group theory and linear algebra. Connell perfectly describes the intentions and style he's shooting for here at the website of the book as follows: The present situation with college textbooks is a national disgrace. Textbooks are too big and too expensive. I was determined to speak right to the student and to have a short book, but keeping it under control was not easy. I worked at it off and on for fourteen years. It came out a little unusual, not by accident or oversight, but because I designed it that way. It is almost entirely the product of my will. Unfortunately mathematics is a difficult and heavy subject. The style and approach of this book is to make it a little lighter. The student has limited time during the semester for serious study, and this time should be allocated with care. The professor picks which topics to assign for serious study and which ones to "wave arms at". The focus is on going forward, because mathematics is learned in hindsight. This book works best when viewed lightly and read as a story, like a graduate student talking to an undergraduate over pizza. Amen. I heartily recommend Connell's book for all students and teachers of algebra as a prototype of what a free open textbook on mathematics should look like. We should all support and encourage such efforts and may there be many more like it. Most highly recommended! Contents Chapter 1 Background and Fundamentals of Mathematics Chapter 2 Groups Chapter 3 Rings Chapter 4 Matrices and Matrix Rings Chapter 5 Linear Algebra Chapter 6 Appendix
- Undergraduate Abstract Algebra Tyler J Evans University of California Davis 2000 (PG)A very terse, concise and somewhat dry set of notes for a first course in algebra emphasizing the interplay of group theory and linear algebra.Proofs are very curt and there are few examples. The notes do not discuss rings, fields or modules-just groups and vector spaces.This wouldn't be so bad in a one semester course-it's doubtful you could cover more in an average paced course. But with the bluntness and lack of detail, the dryness of the notes really turns the reader off.To make matters worse, there's no originality-the notes simply do everything the usual manner with nothing different or inventive. They get the job done and that's about it.You can check them out if you want, but I'd pass-there are so many better sources online for a limited course like this.
- Introduction to Modern Algebra Birne Binegar University of Oklohoma Spring 2013 (PG)Another excellent set of course materials from Binegar, these for a strong first course in abstract algebra. They are of a similar format as his other lecture notes we've seen here, like the ones on linear algebra and advanced calculus. They are concise and focused, but very lucid with an excellent choice of topics and many examples. They are parsimonious with words without lacking the essential details or clarity of explanation. They are particularly strong on foundational topics of logic and basic set theory, for students for whom this will be their first rigorous mathematics course. The Great Debate in beginning abstract algebra courses since Saunders MacLane and Garrett Birkoff taught the first one at Harvard almost three-quarters of a century ago is do you cover rings first or groups first? Each approach has both
- drawbacks and advantages. Most algebra teachers today do groups first and the main advantage is logical order of simplicity. Groups are simpler in structure then rings and most of the basic properties of rings are either derived from or are extensions of those of groups. As I've said before, I'm a groups first man myself. Binegar comes down on the side of the minority-but he develops a great deal of number theory first. This is the traditional way to begin if you want to begin with rings and it makes a lot of sense. The central prototype of a ring is after all, the integers under addition and multiplication. It also allows the examples of rings to be used as examples of Abelian groups. But to be honest, since groups are simpler, I'd tend to think this order would be more confusing for the beginner. Still, these notes are quite readable and relatively brief and there are many good exercises to supplement them. A very solid set of lecture notes and exercises on abstract algebra for he beginner. Highly recommended. Contents I.
- Introduction to Proofs A.Elements of Mathematical Logic B. Methods of Proof C. Review of Set Theory D. Functions II. Arithmetic in Z A. The Division Algorithm B. Divisibility C. Prime Numbers First Midterm III. Modular Arithmetic A. Congruence and Congruence Classes B. Modular Arithmetic C. The Structure of Zp when p is Prime IV. Rings A. Definition and Examples of Rings B. Basic Properties of Rings C. Homorphisms and Isomorphisms of Rings Second Midterm V. The Ring of Polynomials F[x] A. Polynomial Arithmetic and the Division Algorithm B. Divisibility in F[x] C. Irreducible Polynomials and Unique Factorization D. Polynomial Functions, Roots, and Reducibility VI. Groups A. Definition and Examples of Groups B. Basic Properties of Groups C. Subgroups D. Group Homomorphisms
- Undergraduate Group Theory Through The Rubik's Cube Daniel Bump Stanford University 2009 (PG-13) A very inventive and deep set of lecture notes on group theory for undergraduates by an expert on the subject.The course is exactly what the title says it is: an investigation of the symmetry groups in R2
and R3 motivated by the legendary 6 sided object of obsession for mathematicians, computer scientists, students and all around geeks since Ronald Reagan was president. ( A depressing question in relation to my own age: I wonder-how many of his students would have actually heard of a Rubik's Cube before this class? Unless their parents were cubeheads when they were young and continued the interest as parents-for which they'd probably be ostracized as the neighborhood wierdos, not many.) In any event, Bump follows the lead of David Singerman, who was one of the first mathematicians to extensively study the geometric transformations of the Cube in terms of group theory. It turns out, shockingly, most of the concepts of group theory can be implemented in the special cases of the symmetry groups of the Cube-even those which are not geometric. For example, a selective labeling of the various "cells" on each side of the cube can be used to implement modular arithmetic. Other applications of the rigid motions of the cube yield symmetry groups,normal subgroups, group actions and much more. The writing is lively and insightful, with many wonderful pictures as well as abstract proofs. Sadly, there are very few exercises. That being said, Bump has written a wonderful supplement for the group theory section of an algebra course at either the basic or advanced level. I really hope he continues to make it freely available at his website and eventually expands it into a full blown text on group theory. Until than,I most strongly recommend it for all algebra and serious geometry students and teachers.
- Introduction to Modern Algebra David Joyce Clark University 2008(PG-13) A really oddball book draft in abstract algebra that really impressed and bewildered me at the same time.I found it quite by accident, began reading it and couldn't stop until I was halfway through it. It's a book for a fairly high level undergraduate abstract algebra course-definitely one more sophisticated then the usual texts such as Gallian,but not quite as sophisticated as Herstien or Artin. At the same time, category theory is used nearly throughout. But modules and the related sophisticated machinery of commutative algebra and
- tensor products aren't here. And that's just the beginning of how this course breaks the mold. Where most courses begin with either groups or rings, Joyce begins with an overview of algebraic structures in
- general followed by an in-depth presentation of the basics of fields. Except for Howes course, I can't think of any algebra course-online or in print-that does this. In some ways, the order of presentation in these notes is backwards-fields to rings to groups. So you get the idea, this is a rather strange brew. But Joyce knows exactly what he's doing and the result is a shockingly good and innovative text on algebra. There are a stunning number of examples, many standard, but many nonstandard. Some very unusual topics discussed in detail are the dyadic rationals, the Guassian integers, Boolean rings and algebras and
- Krull's theorem. In particular, Joyce gives a wonderful and very elementary treatment of cyclic rings and finite fields. He also covers the construction of the reals via Dedekind cuts from the rationals in very clear and simple manner. There are many nice if brief historical notes interwoven into the larger discussion and they are integrated in a way I've never seen. For example, Joyce uses a detailed discussion of
- Hamilton's discovery of the skew field of quarternions to introduce matrices. There are an enormous number of exercises, none too hard, also interwoven into the chapters themselves rather then at the end of the sections. I have to say I was really surprised how much I liked these notes.I'd love to try and use them one day to teach a course in algebra. I suspect it would be difficult to find a class of students
- they'd be appropriate for in the US-they're be too difficult for average undergraduates and too easy for first year graduate courses. For those looking for a highly unusual and challenging course for honors students, this is definitely worth a look. But I'd be really hesitant to use it with an average class of undergraduate algebra students at any but the very strongest schools.Still-Joyce has written a remarkable, inspiring course and I highly recommend it for all students and teachers of algebra. Contents 1 Introduction 1.1 Structures in Modern Algebra 1.2 Isomorphisms, homomorphisms, etc 1.3 A little number theory 1.4 The fundamental theorem of arithmetic: the unique factorization theorem 2 Fields 2.2 Cyclic rings and finite fields 2.3 Field Extensions, algebraic fields, the complex numbers 2.4 Real numbers and ordered fields 2.5 Skew fields (division rings) and the quaternions 3 Rings 3.1 Introduction to rings 3.2 Factoring Zn by the Chinese remainder theorem 3.3 Boolean rings 3.4 The field of rational numbers and general fields of fractions 3.5 Categories and the category of rings 3.6 Kernels, ideals, and quotient rings 3.7 Krull’s theorem, Zorn’s Lemma, and the Axiom of Choice 3.8 UFDs, PIDs, and EDs 3.9 Polynomial rings 3.10 Rational and integer polynomial rings 4 Groups 4.1 Groups and subgroups 4.2 Symmetric Groups Sn 4.3 Cayley’s theorem and Cayley graphs 4.4 Kernels, normal subgroups, and quotient groups 4.5 Matrix rings and linear groups 4.6 Structure of finite groups 4.7 Abelian groups Index
- Algebraic Structures II M Fayers Queen Mary College University of London 2014 (PG-13)Very solid if standard course in group theory at the undergraduate UK style.Following the traditional mold for such a
- course, the geometric role of symmetry groups and group actions as Euclidean transformations are emphasized. There are many careful yet concise proofs, as well as many examples and pictures. It's pitched at a somewhat higher level then most US courses in algebra and presumes a good working knowledge of both linear algebra and the elements of ring theory. Very readable with many good problems. The material is very standard, but who cares? A basic course doesn't have to be innovative to
- be well done-and this is well done. Highly recommended for strong undergraduates or teachers of algebra. Contents: Group homomorphisms, isomorphisms, automorphisms. Conjugacy and normal subgroups; examples.Construction of factor groups, 1st, 2nd and 3rd isomorphism theorems for groups; examples.Group actions; finite p-groups; Sylow theorems and applications.Jordan–Hölder theorem; finite soluble groups; examples.
- Algebraic Structures I Peter J. Cameron King's College University of London 2005(PG-13)
- Ten Chapters of the Algebraical Art Peter J. Cameron University of London 2007(PG) The persona of Peter Cameron now haunts most of the algebra courses at Queen Mary, as his excellent textbook Introduction To Algebra, is now in it's second edition and gaining popularity in the UK as a basic algebra text.Fortunately for the entire online mathematical community, Cameron has very graciously allowed his original lecture notes for 2 versions of his courses taught there to remain freely available at his website. Indeed, these notes were the basis for his rewriting the first edition of the book into the current edition!As one would expect, there's considerable overlap between the two sets of notes, despite being written for 2 different levels of algebra course.The first link is to the standard second or third year undergraduate algebra course for which Cameron wrote the first edition of the book-and began to draft the second edition from.The course presumes the students have taken both a basic linear algebra course and a foundational course in rigorous mathematics( logic, naive set theory and the machinery of proofs) in the UK style. Cameron is a "rings first" man as far as presenting order is concerned and I've commented on the pros and cons of the rival approaches to teaching algebra elsewhere at this site.That being said, Cameron writes both concisely and clearly, something a lot of mathematics textbook authors have problems doing simultaneously. He particularly does a very nice job developing basic ring theory for
- beginners.He makes a real effort to not assume readers will "get" foundational details, such as summation notation or permutation cycle notation-so he's extremely detailed in both specific definitions and notations. Beginners will appreciate this- especially for little technicalities that aren't obvious. For example, like a true algebraicist, he writes functions on the right.and Cameron is very cautious to stop and warn the student. (Which I wish my honors algebra instructor had done about Herstien.......) There are many good if not overly creative examples. For the most part, Cameron is intentionally trying to be very standard here and isn't trying to be creative. One very good creative touch that he does add to both sets of notes are a number of interesting historical notes, many I wasn't familiar with. The other course, the 2007 course, has a great deal of overlap with the 2005 course in the little details, but it's organized and contented quite differently. This course-despite the title-is intended actually for the foundational course which is assumed as prerequisite in the 2005 course! It has no clear prerequisites and covers mostly the elements of rigorous mathematics and linear algebra the previous course assumes. As a result,
- the actual abstract algebra material is dramatically scaled down. As I've said, though-Cameron writes very well and covers all this material in a readable manner targeted to beginners. I'd recommend for those attempting to use them for self-study, use them in reverse chronological order-this is the order Cameron intended for them to be used as course texts. Unfortunately, there's a huge problem trying to use them as course texts- there are no exercises. Of course, the exercises are intended to come from his book. For students who'd like to use these notes for self study, if you can borrow a copy and photocopy all the exercises, that would be ideal. Otherwise, they'll make fine supplementary material for sources that have exercises elsewhere. Highly recommended for both teachers and students of beginning abstract
- algebra. Contents: (Algebraic Structures) Contents Introduction 1.1 Abstract algebra 1.2 Sets, functions, relations 1.3 Equivalence relations and partitions 1.4 Matrices 1.5 Polynomials 1.6 Permutations 2
- Rings 2.1 Introduction 2.1.1 Definition of a ring 2.1.2 Examples of rings 2.1.3 Properties of rings 2.1.4 Matrix rings 2.1.5 Polynomial rings 2.2 Subrings 2.2.1 Definition and test 2.2.2 Cosets 2.3
- Homomorphisms and quotient rings 2.3.1 Isomorphism 2.3.2 Homomorphisms 2.3.3 Ideals 2.3.4 Quotient rings 2.3.5 The Isomorphism Theorems 2.4 Factorisation 2.4.1 Zero divisors and units 2.4.2 Unique factorisation domains 2.4.3 Principal ideal domains 2.4.4 Euclidean domains 2.4.5
- Appendix 2.5 Fields 2.5.1 Maximal ideals 2.5.2 Adding the root of a polynomial 2.5.3 Finite fields 2.5.4 Field of fractions 2.5.5 Appendix: Simple rings 2.5.6 Appendix: The number systems 3 Groups 3.1
- Introduction 3.1.1 Definition of a group 3.1.2 Examples of groups 3.1.3 Properties of groups 3.1.4 Notation 3.1.5 Order 3.1.6 Symmetric groups 3.2 Subgroups 3.2.1 Subgroups and subgroup tests 3.2.2 Cyclic groups 3.2.3 Cosets 3.2.4 Lagrange’s Theorem 3.3 Homomorphisms and normal subgroups 3.3.1 Isomorphism 3.3.2 Homomorphisms 3.3.3 Normal subgroups 3.3.4 Quotient groups 3.3.5 The Isomorphism Theorems 3.3.6 Conjugacy 3.4 Symmetric groups and Cayley’s Theorem 3.4.1 Proof of Cayley’s Theorem 3.4.2 Conjugacy in symmetric groups 3.4.3 The alternating groups 3.5 Some special groups 3.5.1 Normal subgroups of S4 and S5 3.5.2 Dihedral groups 3.5.3 Small groups 3.5.4 Polyhedral groups (Ten Chapters) Contents 1 What is mathematics about? 2 Numbers 3 Other algebraic systems 2 4 Relations and functions 5 Division and Euclid’s algorithm 6 Modular arithmetic 7 Polynomials revisited 8 Rings 9 Groups 10 Permutations
- Groups, Rings and Modules .J.B.Brookes University of Cambridge Lent 2005 (PG-13)Yet another set of notes for the basic undergraduate algebra sequence at Cambridge.You know the spiel by now: Concise (although a bit more detailed then the usual Cambridge notes) , clear and well written with no exercises, and a bit more sophisticated then the usual American first course.That being said, the standard material is presented very lucidly and relatively briefly.There are many good examples and several unusual results, such as the proof of the fact that the only simple abelian groups are Cp for p a prime integer.The student is assumed to have mastered the material of the introductory mathematics
- course Algebra And Geometry at Cambridge, which means the student is supposed to have a good grasp of naive set theory, basic proof skills and the elements of linear algebra.There are better sources available, but a serious student or teacher of algebra will find these notes quite accessible and useful as a supplementary text. Recommended.
- A GENTLE INTRODUCTION TO ABSTRACT ALGEBRA by B.A. Sethuraman California State University
Northridge (PG) Another free of charge, open textbook on abstract algebra for the beginning student-hooray!As I've said, the more of these kinds of texts that are available, all the better for the plebian independent scholars. We've already seen several excellent ones-Judson, Goodman, Connell.Is this
- another quality one? Indeed it is. The author's introduction to the student, "How to Read a Mathematics", by itself, would be worth downloading as a standalone article for students beginning serious mathematics. But then one would miss out on the rest of this very solid, extremely readable text. The title is very appropriate for the book. The entire book is geared to present abstract algebra in a form palatable for students with a background of only practical calculus and some linear algebra. This is one of the reasons the book begins with rings and fields- they can naturally be generalized from number systems. At least, that's the thinking behind it. I've commented on this elsewhere and won't repeat it here. In this case, it works well since the author introduces groups and rings simultaneously. While the book does strongly encourage the reader to actively read with pencil in hand, the full consequences of the definitions and theorems are slowly unfurled to the reader in tiny bites. There is a very large number of
- examples,mostly standard, but presented in greater detail then usual. Subtle details of both definitions and proofs of major results are spelled out in pedantic, almost excruciating detail-which is quite
- helpful to the absolute beginner. Many of these details are shunted to very large collection of exercises, many of which are woven into the text proper in addition to end of the chapter problems and supplementary challenge problems. Many of the in-text problems-called "Questions"- are thought exercises designed to strengthen and clarify the student's understanding of basic concepts. For example, in the chapter on the integers, Sethuraman asks whether or not an infinite set of integers has a maximal element and if so, why. Students have to understand not only the ordering properties of the integers to answer this question, but how those properties determine the large scale results such as induction.
- The book is peppered with problems like this and they certainly assist the student in learning how to think mathematically. It's clear this is a book really designed first and foremost to teach beginners algebra and it does so quite well. Another excellent online book on algebra for beginners. Highly recommended. Contents Preface To the Student: How to Read a Mathematics Book 1 Divisibility in the Integers 2 Rings and Fields 2.1 Rings: Definition and Examples 2.2 Subrings 2.3 Integral Domains and Fields 2.4 Ideals 2.5 Quotient Rings 2.6 Ring Homomorphisms and Isomorphisms 2.7 Further Exercises 3 Vector Spaces 3.1 Vector Spaces: Definition and Examples 3.2 Linear Independence, Bases,
- Dimension 3.3 Subspaces and Quotient Spaces 3.4 Vector Space Homomorphisms: Linear Transformations 3.5 Further Exercises 4 Groups 4.1 Groups: Definition and Examples 4.2 Subgroups, Cosets, Lagrange's Theorem 4.3 Normal Subgroups, Quotient Groups 4.4 Group Homomorphisms
- and Isomorphisms 4.5 Further Exercises A Sets, Functions, and Relations B Partially Ordered Sets, Zorn's Lemma C GNU Free Documentation License GNU Free Documentation License
- Notes on Abstract Algebra: Scott M. LaLonde Dartmouth University Summer 2013 (PG) A very intense if standard course in abstract algebra for a relatively short summer course,focusing on groups and rings.It always amazes me that universities actually offer an undergraduate abstract algebra course in the summer semesters, which strikes me as a fairly insane thing to do.What's even more amazing to me is that students actually willingly sign up and take itin the summertime. I remember basic calculus being a nightmare in the sweltering heat of July and early August, covering roughly what would be 3
weeks worth of material, in a normal fall or spring pace, in a single 4 hour day with 20 minutes for lunch.I can't imagine taking an actual rigorous mathematics course-especially one as intense as abstract
- algebra-under those conditions. The only students I can see doing this are the true hard core stars of the honors cadre in the department- students who commit to eat,breathe and sleep mathematics, don't work and are determined to be taking graduate courses as juniors before applying to MIT and Yale for graduate school. Be that as it may, LaLonde does a very nice job trying to compose a reasonable course that can actually be completed by dedicated students in the limited time frame..I strongly suspect the author crafted these notes in the months preceding the course and assigned large chunks of it to the class for assigned reading and homework-it's hard to imagine the class being able to cover the bulk of the notes in the allotted time otherwise! Readablity of the notes is paramount in a situation like this and the
- author gets high marks for that. He's able to be relatively detailed and he does this by being very selective in his coverage. He covers basic number theory, group theory, ring theory and the very barest elements of
- Galois theory on polynomials-and that's it. Period. There's really only enough material here for a one semester first course-anyone who wants to use it for self study in algebra for a whole year course is going to have to augment it with material on field theory, R-modules and other topics. Still, given the enormous time pressure of the schedule, he covers a surprisingly large amount of material.There are many examples, mostly standard, but well presented), nice historical notes to enliven the presentation further and he gives some interesting footnotes that point the reader towards more subtle points and further reading. A very solid first course in abstract algebra for mathematics majors. Highly recommended. Contents 1 Introduction 1.1 What is Abstract Algebra? 1.1.1 History 1.1.2 Abstraction 1.2 Motivating Examples 1.2.1 The Integers 1.2.2 Matrices 1.3 The integers mod n 1.3.1 The Euclidean Algorithm 2 Group Theory 2.1 Definitions and Examples of Groups 2.1.1 Binary Operations 2.1.2 Groups 2.1.3 Group Tables 2.1.4 Remarks on Notation 2.2 The Symmetric and Dihedral Groups 2.2.1 The Symmetric Group
- 2.2.2 The Dihedral Group 2.3 Basic Properties of Groups 2.4 The Order of an Element and Cyclic Groups 2.4.1 Cyclic Groups 2.4.2 Classication of Cyclic Groups 2.5 Subgroups 2.5.1 Cyclic Subgroups 2.5.2 Subgroup Criteria 2.5.3 Subgroups of Cylic Groups 2.6 Lagrange's Theorem 2.6.1 Equivalence Relations 2.6.2 Cosets 2.7 Homomorphisms 2.7.1 Basic Properties of Homomorphisms 2.8 The Symmetric Group Redux 2.8.1 Cycle Decomposition 2.8.2 Application to Dihedral Groups 2.8.3 Cayley's Theorem 2.8.4 Even and Odd Permutations and the Alternating Group 2.9 Kernels of Homomorphisms 2.10 Quotient Groups and Normal Subgroups 2.10.1 The Integers mod n 2.10.2 General Quotient Groups 2.10.3 Normal
- Subgroups 2.10.4 The First Isomorphism Theorem 2.10.5 Aside: Applications of Quotient Groups 2.11 Direct Products of Groups 2.12 The Classication of Finite Abelian Groups 3 Ring Theory 3.1 Rings 3.2 Basic Facts and Properties of Rings 3.2.1 The Quaternions 3.3 Ring Homomorphisms and Ideals 3.4 Quotient Rings 3.5 Polynomials and Galois Theory 3.6 Act I: Roots of Polynomials 3.7 Act II: Field Extensions 3.8 Act III: Galois Theory 3.8.1 Epilogue A Set Theory A.1 Sets A.2 Constructions on Sets A.3 Set Functions A.4 Notation B Techniques for Proof Writing B.1 Basic Proof Writing B.2 Proof by Contradiction B.3 Mathematical Induction B.4 Proof by Contrapositive B.5 Tips and Tricks for Proofs
- Introduction to Abstract Algebra Samir Siksek Mathematics Institute University of Warwick 2013(PG)A strong, readable and surprisingly funny set of notes for a one semester first course in abstract algebra.
- Interestingly, the selection and ordering of topics is very similar to LeMonde's lecture notes we just reviewed.The main difference is that Siksek assumes somewhat less background and covers basic linear
- algebra in somewhat more depth in his notes-although he does assign the linear algebra portions as independent reading.Siksek lays out a fairly standard one semester algebra course, but it's deeper and more insightful then the usual such course Like a lot of mathematics courses from the UK, the quality lies in the focus on the little details that American courses omit. For example, Siksek gives a wonderful geometric explanation, using Euclidean transformations of the plane, of why matrix multiplication is defined the way it is and why it's not commutative. He also discusses at some length the definition of matrices as linear maps operating on row vectors. He also gives a very detailed and readable presentation of symmetry groups in classical geometry.The big surprise here is how amusing the notes can be in places. Humor,sadly, isn't something we see often in lecture notes and textbooks on university level mathematics, but these have it in abundance.A couple of examples for you edification: On the title page: 101% free of subliminal messages. Page 13: You see, even though the quaternions have been consigned to the compost heap of algebra,Hamilton’s graffiti became history’s most celebrated act of mathematical vandalism. There is a great moral to this, but I can’t find it. In short , these are very solid and informative set of notes that don't try to do too much and manage to be as entertaining as they
- are educational. They'll make a fine resource for a first course in abstract algebra. Highly recommended!
Contents Chapter I. Prologue I.1. Who Am I? I.2. A Jolly Good Read! I.3. Proofs I.4. Acknowledgements and Corrections Chapter II. FAQ Chapter III. Algebraic Reorientation III.1. Sets III.2. Binary Operations III.3. Vector Operations III.4. Operations on Polynomials III.5. Composition of Functions III.6. Composition Tables III.7. Commutativity and Associativity III.8. Where are the Proofs? III.9. The Quaternionic Number System (do not read) Chapter IV. Matrices—Read On Your Own IV.1. What are Matrices? 15 IV.2. Matrix Operations IV.3. Where do matrices come from? IV.4. How to think
- About matrices? IV.5. Why Column Vectors? IV.6. Multiplicative Identity and Multiplicative Inverse IV.7. Rotations Chapter V. Groups V.1. The Definition of a Group V.2. First Examples (and Non-Examples) V.3. Abelian Groups V.4. Symmetries of a Square Chapter VI. First Theorems VI.1. Getting Relaxed about Notation VI.2. Additive Notation Chapter VII. More Examples of Groups VII.1. Matrix Groups I VII.2. Congruence Classes Chapter VIII. Orders and Lagrange’s Theorem VIII.1. The Order of an Element VIII.2. Lagrange’s Theorem—Version 1 Chapter IX. Subgroups IX.1. What Were They Again? IX.2. Criterion for a Subgroup IX.3. Roots of Unity IX.4. Matrix Groups II IX.5. Differential Equations
- IX.6. Non-Trivial and Proper Subgroups IX.7. Lagrange’s Theorem—Version 2 Chapter X. Cyclic Groups and Cyclic Subgroups X.1. Lagrange Revisited X.2. Subgroups of Z Chapter XI. Isomorphisms Chapter XII. Cosets XII.1. Geometric Examples XII.2. Solving Equations XII.3. Index XII.4. The First Innermost Secret of Cosets XII.5. The Second Innermost Secret of Cosets XII.6. Lagrange Super-Strength Chapter XIII. Quotient Groups XIII.1. Congruences Modulo Subgroups XIII.2. Congruence Classes and Cosets XIII.3. R/Z XIII.4. R2/Z2 XIII.5. R/Q XIII.6. Well-Defined and Proofs Chapter XIV. Symmetric Groups XIV.1. Motivation XIV.2. Injections, Surjections and Bijections XIV.3. The Symmetric Group XIV.4. Sn XIV.5. A Nice Application of Lagrange’s Theorem XIV.6. Cycle Notation XIV.7. Permutations and Transpositions XIV.8. Even and Odd Permutations Chapter XV. Rings XV.1. Definition XV.2. Examples XV.3. Subrings XV.4. The Unit Group of a Ring XV.5. The Unit Group of the Gaussian Integers Chapter XVI. Fields Chapter XVII. Congruences Revisited XVII.1. Units in Z/mZ XVII.2. Fermat’s Little Theorem XVII.3. Euler’s Theorem XVII.4. Vale Dicere Appendices Appendix A. 2012 Introduction to Abstract Algebra Paper Appendix B. 2013 Introduction to Abstract Algebra Paper Appendix C. The Forgotten Joys of Analytic Irresponsibility C.1. The Mathematical Equivalent of an X-Rated DVD C.2. Nothing to see here—move along please