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Essential Group Theory PDF

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Essential Group Theory Michael Batty Download free books at Michael Batty Essential Group Theory Download free eBooks at bookboon.com 2 Essential Group Theory © 2012 Michael Batty & bookboon.com ISBN 978-87-403-0301-8 Download free eBooks at bookboon.com 3 Essential Group Theory Contents Contents Introduction 9 1 Sets and Maps 10 1.1 Sets 10 1.2 Maps 11 1.3 Equivalence Relations and Partitions 12 1.4 Modular Arithmetic 13 2 Groups 14 2.1 Binary Operations 14 2.2 Groups: Basic Definitions 15 2.3 Examples of Groups 17 3 Subgroups 21 3.1 Definition of a Subgroup 21 3.2 Cosets 22 3.3 Lagrange’s Theorem 23 III jjjoooiiinnneeeddd MMMIIITTTAAASSS bbbeeecccaaauuussseee ���eee GGGrrraaaddduuuaaattteee PPPrrrooogggrrraaammmmmmeee fffooorrr EEEnnngggiiinnneeeeeerrrsss aaannnddd GGGeeeooosssccciiieeennntttiiissstttsss III wwwaaannnttteeeddd rrreeeaaalll rrreeessspppooonnnsssiiibbbiiillliii��� wwwMMM.daaaiseeecrrrosssvkkke...cccrmooommmita///MMMs.iiictttoaaamsss I joined MITAS because �e Graduate Programme for Engineers and Geoscientists I wanted real responsibili� Maersk.com/Mitas MMMooonnnttthhh 111666 IIIIII wwwwwwaaaaaassssss aaaaaa cccooonnnssstttrrruuuccctttiiiooonnn Month 16 sssuuupppeeerrrvvviiiIIsss wwooorrraa iiissnnn aa construction ttthhheee NNNooorrrttthhh SSSeeeaaa supervisor in aaadddvvviiisssiiinnnggg aaannnddd the North Sea hhhhhheeeeeelllpppiiinnnggg fffooorrreeemmmeeennn advising and RRReeeaaalll wwwooorrrkkk IIIIIInnnnnntttttteeeeeerrrrrrnnnnnnaaaaaattttttiiiiiioooooonnnnnnaaaaaaaaallllll oooppppppooorrrtttuuunnniiitttiiieeesss ssssssooolllvvveee ppprrrooobbbllleeemmmhhssseelping foremen ������rrrrrreeeeeeeeeeee wwwwwwooooooooorrrrrrkkk ppplllaaaccceeemmmeeennntttsss Real work IInntteerrnnaattiioonnaaall opportunities ssolve problems ��rreeee wwooorrk placements Download free eBooks at bookboon.com 4 Click on the ad to read more Essential Group Theory Contents 4 Generators and Cyclic Groups 24 4.1 Orders of Group Elements 24 4.2 Generating Sets 25 4.3 Cyclic Groups 26 4.4 Fermat’s Little Theorem 27 5 Mappings of Groups 28 5.1 Homomorphisms 28 5.2 Isomorphisms 29 6 Normal Subgroups 31 6.1 Conjugates and Normal Subgroups 31 6.2 Cosets of Normal Subgroups 32 6.3 Kernels of Homomorphisms 32 7 Quotient Groups 35 7.1 Products of Cosets 35 7.2 Quotient Groups 35 8 The First Isomorphism Theorem 37 8.1 The First Isomorphism Theorem 37 www.job.oticon.dk Download free eBooks at bookboon.com 5 Click on the ad to read more Essential Group Theory Contents 8.2 Centres and Inner Automorphisms 39 9 Group Actions 40 9.1 Actions of Groups 40 9.2 The Orbit-Stabilizer Theorem 41 10 Direct Products 43 10.1 Direct Products 43 10.2 Direct Products of Finite Cyclic Groups 43 10.3 Properties of Direct Products 44 11 Sylow Theory 47 11.1 Primes and p-Groups 47 11.2 Sylow’s Theorem 48 12 Presentations of Groups 51 12.1 Introduction to Presentations 51 12.2 Alphabets and Words 53 12.3 Von Dyck’s Theorem 56 12.4 Finitely Generated and Finitely Presented Groups 57 12.5 Dehn’s Fundamental Algorithmic Problems 58 Study at Linköping University and get the competitive edge! Interested in Computer Science and connected fields? Kick-start your career with a master’s degree from Linköping University, Sweden – one of top 50 universities under 50 years old. www.liu.se/master Download free eBooks at bookboon.com 6 Click on the ad to read more Essential Group Theory Contents 13 Free Groups 60 13.1 Reduced Words and Free Groups 60 13.2 Normal Closure 61 13.3 Torsion Free Groups 62 14 Abelian Groups 64 14.1 Commutator Subgroups and Abelianisations 64 14.2 Free Abelian Groups 65 14.3 Finitely Generated Abelian Groups 67 14.4 Generalisations of Abelian Groups 67 15 Transforming Presentations 69 15.1 Tietze Transformations 69 15.2 Properties of Tietze Transformations 71 16 Free Products 74 16.1 Free Products 74 16.2 A Normal Form for Free Products 76 16.3 The Universal Property of Free Products 77 16.4 Independence of Presentation 78 16.5 Decomposability 78 (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:2)(cid:8)(cid:9)(cid:10)(cid:2)(cid:7)(cid:9)(cid:5)(cid:10)(cid:2)(cid:11)(cid:12)(cid:3)(cid:4)(cid:13)(cid:10)(cid:14)(cid:3)(cid:2)(cid:6)(cid:13)(cid:15)(cid:10)(cid:13)(cid:13)(cid:2) (cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:7)(cid:8)(cid:9)(cid:13)(cid:15)(cid:16)(cid:10)(cid:17)(cid:18)(cid:10)(cid:19)(cid:7)(cid:20)(cid:3)(cid:12)(cid:17)(cid:5)(cid:17)(cid:21)(cid:16)(cid:10)(cid:20)(cid:17)(cid:12)(cid:22)(cid:23)(cid:20)(cid:15)(cid:9)(cid:10)(cid:8)(cid:7)(cid:9)(cid:7)(cid:4)(cid:8)(cid:20)(cid:3)(cid:10)(cid:4)(cid:12)(cid:22)(cid:10)(cid:7)(cid:22)(cid:23)(cid:20)(cid:4)(cid:15)(cid:13)(cid:17)(cid:12)(cid:10)(cid:13)(cid:12)(cid:10)(cid:7)(cid:12)(cid:21)(cid:13)(cid:12)(cid:7)(cid:7)(cid:8)(cid:24) (cid:13)(cid:12)(cid:21)(cid:10)(cid:4)(cid:12)(cid:22)(cid:10)(cid:12)(cid:4)(cid:15)(cid:23)(cid:8)(cid:4)(cid:5)(cid:10)(cid:9)(cid:20)(cid:13)(cid:7)(cid:12)(cid:20)(cid:7)(cid:9)(cid:25)(cid:10)(cid:4)(cid:8)(cid:20)(cid:3)(cid:13)(cid:15)(cid:7)(cid:20)(cid:15)(cid:23)(cid:8)(cid:7)(cid:25)(cid:10)(cid:15)(cid:7)(cid:20)(cid:3)(cid:12)(cid:17)(cid:5)(cid:17)(cid:21)(cid:16)(cid:24)(cid:8)(cid:7)(cid:5)(cid:4)(cid:15)(cid:7)(cid:22)(cid:10)(cid:6)(cid:4)(cid:15)(cid:3)(cid:7)(cid:6)(cid:4)(cid:15)(cid:13)(cid:20)(cid:4)(cid:5)(cid:10)(cid:9)(cid:20)(cid:13)(cid:7)(cid:12)(cid:20)(cid:7)(cid:9)(cid:10) (cid:4)(cid:12)(cid:22)(cid:10)(cid:12)(cid:4)(cid:23)(cid:15)(cid:13)(cid:20)(cid:4)(cid:5)(cid:10)(cid:9)(cid:20)(cid:13)(cid:7)(cid:12)(cid:20)(cid:7)(cid:9)(cid:26)(cid:10)(cid:27)(cid:7)(cid:3)(cid:13)(cid:12)(cid:22)(cid:10)(cid:4)(cid:5)(cid:5)(cid:10)(cid:15)(cid:3)(cid:4)(cid:15)(cid:10)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:4)(cid:20)(cid:20)(cid:17)(cid:6)(cid:28)(cid:5)(cid:13)(cid:9)(cid:3)(cid:7)(cid:9)(cid:25)(cid:10)(cid:15)(cid:3)(cid:7)(cid:10)(cid:4)(cid:13)(cid:6)(cid:10)(cid:28)(cid:7)(cid:8)(cid:9)(cid:13)(cid:9)(cid:15)(cid:9)(cid:10) (cid:18)(cid:17)(cid:8)(cid:10)(cid:20)(cid:17)(cid:12)(cid:15)(cid:8)(cid:13)(cid:29)(cid:23)(cid:15)(cid:13)(cid:12)(cid:21)(cid:10)(cid:15)(cid:17)(cid:10)(cid:4)(cid:10)(cid:9)(cid:23)(cid:9)(cid:15)(cid:4)(cid:13)(cid:12)(cid:4)(cid:29)(cid:5)(cid:7)(cid:10)(cid:18)(cid:23)(cid:15)(cid:23)(cid:8)(cid:7)(cid:10)(cid:30)(cid:10)(cid:29)(cid:17)(cid:15)(cid:3)(cid:10)(cid:12)(cid:4)(cid:15)(cid:13)(cid:17)(cid:12)(cid:4)(cid:5)(cid:5)(cid:16)(cid:10)(cid:4)(cid:12)(cid:22)(cid:10)(cid:21)(cid:5)(cid:17)(cid:29)(cid:4)(cid:5)(cid:5)(cid:16)(cid:26) (cid:31)(cid:13)(cid:9)(cid:13)(cid:15)(cid:10)(cid:23)(cid:9)(cid:10)(cid:17)(cid:12)(cid:10)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:9)(cid:7)(cid:10)(cid:17)(cid:8)(cid:10)(cid:11)(cid:7)(cid:12)(cid:13)(cid:14)(cid:15)(cid:13)(cid:16)(cid:17)(cid:14)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:17)(cid:12)(cid:10)(cid:18)(cid:4)(cid:20)(cid:7)(cid:29)(cid:17)(cid:17) (cid:26)(cid:10) Download free eBooks at bookboon.com 7 Click on the ad to read more Essential Group Theory Contents 17 F ree Products With Amalgamation 80 17.1 Free Products with Amalgamation 80 17.2 Pushouts 81 17.3 Independence of Presentation 85 18 HNN Extensions 86 18.1 HNN Extensions 86 18.2 Relation to Free Products with Amalgamation 87 18.3 The Higman-Neumann-Neumann Embedding Theorem 90 19 Further Reading 92 20 Bibliography 94 21 Index 95 welcome to our world of teaching! innovation, flat hierarchies and open-Minded professors Study in Sweden - cloSe collaboration with future employerS Mälardalen university collaborates with Many eMployers such as abb, volvo and ericsson debajyoti nag taKe the sweden, and particularly Mdh, has a very iMpres- sive reputation in the field of eMbedded systeMs re- right tracK search, and the course design is very close to the industry requireMents. he’ll tell you all about it and answer your questions at give your career a headStart at mälardalen univerSity mduStudent.com www.mdh.se Download free eBooks at bookboon.com 8 Click on the ad to read more Essential Group Theory Introduction Introduction This short book on group theory is partly based on notes from lectures I gave at Trinity College Dublin in 1999 and at Edinburgh University in 2001. The first part of the book is an introduction to elementary group theory. It doesn’t aim to cover everything that might be in your introductory course in abstract algebra, but just to give a summary of the important points. Perhaps you could read it before you begin your course, if you want to read something gentle in advance, or it could be a starting point for revision. The second part of the book is about free groups and presentations of groups. This would typically appear in a second course on group theory. Group theory can seem very abstract and strange when you first encounter it. It involves a different mindset and most likely you will not have done this type of mathematics before. But it has a set of techniques and beauty of its own and is worth perservering with, and you will find that the Gs and Hs will soon come to life. As with a lot of university mathematics, it depends very much on the language and logic of sets and their elements, containment, equality and maps and unless you understand these fully, it is unlikely that you will be able to apply them in the context of group theory, where there is even more to think about. Most of the elementary proofs in group theory involve these simple but important techniques. I can’t stress this enough and have included a preliminary section on sets and maps before the main theme of group theory starts. I hope that there are as few mistakes as possible, but if you find any, have suggestions to improve the book or feel that I have not covered something which should be included please send an email to me at [email protected] Michael Batty, Durham, 2012. Download free eBooks at bookboon.com 9 Essential Group Theory Sets and Maps 1 Sets and Maps This section is primarily for reference, as you will probably have seen most of these definitions before. But please at least skim lightly through them as a reminder and refer to them later when necessary. 1.1 Sets Sets are at the very foundation of mathematics. They are difficult to define formally, in order to avoid things going wrong. This can be done, with various systems of axioms. But it’s a subject in its own right. Better for now if we just naively think of them as collections of elements, and take that as a starting point. We will liberally use set notation throughout the book, summarized as follows: • x X means x is an element of the set X. ∈ • x / X means x is not an element of the set X. ∈ • x,y means the set consisting of x and y. { } • X Y or Y X means all the elements of X are in Y. We say X is a subset of Y or Y ⊂ ⊃ contains X. • X =Y means X Y and Y X. This is how sets are proved to be equal, so such proofs ⊂ ⊂ have two parts. • X Y will also include the possibility that X =Y. We will not use notation like X Y. ⊂ ⊆ • X Y means the union of X and Y, the set of elements that are in X or Y (or both). ∪ • X Y means the intersection of X and Y, the set of elements that are in X and Y. ∩ • X Y means the set of elements that are in X but not in Y. − • denotes the set containing no elements. ∅ {} A special type of set is a pair which has two elements. This can be either unordered, in which case it is of the form x,y { } Download free eBooks at bookboon.com 10

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