Group Theory J.S. Milne S3 Â1 2 3à r D 2 3 1 Â1 2 3à f D 1 3 2 Version 4.00 June 23, 2021 The first version of these notes was written for a first-year graduate algebra course.Asin most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or Lie groups, and it is not just the groups themselves that are of interest, but also their linear representations. It is my intention (one day) to expand the notes to take account of this, and to produce a volume that, while still modest in size (c200 pages), will provide a more comprehensive introduction to group theory for beginning graduate students in mathematics, physics, and related fields. BibTeX information @misc{milneGT, author={Milne, James S.}, title={Group Theory (v4.00)}, year={2021}, note={Available at www.jmilne.org/math/}, pages={139} } Please send comments and corrections to me at jmilne at umich dot edu. v2.01 (August 21, 1996). First version on the web; 57 pages. v2.11 (August 29, 2003). Revised and expanded; numbering; unchanged; 85 pages. v3.00 (September 1, 2007). Revised and expanded; 121 pages. v3.16 (July 16, 2020). Revised and expanded; 137 pages. v4.00 (June 23, 2021). Made document (including source code) available under Creative Commons licence. The multiplication table of S3 on the front page was produced by Group Explorer. Version 4.0 is published under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International licence (CC BY-NC-SA 4.0). Licence information: https://creativecommons.org/licenses/by-nc-sa/4.0 Copyright ©1996–2021 J.S. Milne. Contents Contents3 1 Basic Definitions and Results7 Definitions and examples.............................. 7 Multiplication tables ................................ 11 Subgroups...................................... 12 Groups of small order................................ 15 Homomorphisms .................................. 16 Cosets........................................ 17 Normal subgroups.................................. 18 Kernels and quotients................................ 20 Theorems concerning homomorphisms....................... 21 Direct products ................................... 23 Commutative groups ................................ 25 The order of ab ................................... 28 Exercises ...................................... 29 2 Free Groups and Presentations; Coxeter Groups 31 Free monoids .................................... 31 Free groups ..................................... 32 Generators and relations .............................. 35 Finitely presented groups.............................. 37 Coxeter groups ................................... 38 Exercises ...................................... 41 3 Automorphisms and Extensions 43 Automorphisms of groups ............................. 43 Characteristic subgroups .............................. 45 Semidirect products................................. 46 Extensions of groups ................................ 50 The Holder¨ program. ................................ 52 Exercises ...................................... 54 4 Groups Acting on Sets 57 Definition and examples .............................. 57 Permutation groups................................. 64 The Todd-Coxeter algorithm............................. 70 Primitive actions. .................................. 71 Exercises ...................................... 73 5 The Sylow Theorems; Applications 75 3 The Sylow theorems ................................ 75 Alternative approach to the Sylow theorems.................... 79 Examples ...................................... 80 Exercises ...................................... 83 6 Subnormal Series; Solvable and Nilpotent Groups 85 Subnormal Series................................... 85 Solvable groups................................... 88 Nilpotent groups .................................. 91 Groups with operators................................ 94 Krull-Schmidt theorem............................... 96 Exercises ...................................... 97 7 Representations of Finite Groups 99 Matrix representations ............................... 99 Roots of 1 in fields .................................100 Linear representations................................100 Maschke’s theorem.................................101 The group algebra; semisimplicity .........................103 Semisimple modules ................................104 Simple F -algebras and their modules........................106 Semisimple F -algebras and their modules.....................110 The representations of G ..............................112 The characters of G .................................113 The character table of a group ...........................116 Examples ......................................116 Exercises ......................................116 A Additional Exercises 119 B Solutions to the Exercises 123 C Two-Hour Examination 131 Bibliography 133 Index 135 4 NOTATION. We use the standard (Bourbaki) notation: N 0;1;2;::: ; Z is the ring of integers; Q is the D f g field of rational numbers; R is the field of real numbers; C is the field of complex numbers; Fq is a finite field with q elements, where q is a power of a prime number. In particular, Fp Z=pZ for p a prime number. D For integers m and n, m n means that m divides n, i.e., n mZ. Throughout the notes, j 2 p is a prime number, i.e., p 2;3;5;7;11;:::;1000000007;:::. D Given an equivalence relation, Œ denotes the equivalence class containing . The empty set is denoted by . The cardinality of a set S is denoted by S (so S is the number of ; j j j j elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by 1 I , denoted .ai /i I , is a function i ai I A. 2 7! W ! Rings are required to have an identity element 1, and homomorphisms of rings are required to take 1 to 1. An element a of a ring is a unit if it has an inverse (element b such that ab 1 ba). The identity element of a ring is required to act as 1 on a module over D D the ring. X YX is a subset of Y (not necessarily proper); X def YX is defined to be Y , or equals Y by definition; D X YX is isomorphic to Y ; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism); ' PREREQUISITES An undergraduate “abstract algebra” course. COMPUTER ALGEBRA PROGRAMS GAP is an open source computer algebra program, emphasizing computational group theory. To get started with GAP, I recommend going to Alexander Hulpke’s page here, where you will find versions of GAP for both Windows and Macs and a guide “Abstract Algebra in GAP”. The Sage page here provides a front end for GAP and other programs. I also recommend N. Carter’s “Group Explorer” here for exploring the structure of groups of small order. Earlier versions of these notes (v3.02) described how to use Maple for computations in group theory. ACKNOWLEDGEMENTS I thank the following for providing corrections and comments for earlier versions of these notes: V.V. Acharya; Yunghyun Ahn; Max Black; Tony Bruguier; Vigen Chaltikyan; Dustin Clausen; Benoˆıt Claudon; Keith Conrad; Demetres Christofides; Adam Glesser; Darij Grinberg; Sylvan Jacques; Martin Klazar; Thomas Lamm; Mark Meckes; Max Menzies; Victor Petrov; Flavio Poletti; Diego Silvera; Efthymios Sofos; Dave Simpson; David Speyer; Kenneth Tam; Robert Thompson; Bhupendra Nath Tiwari; Leandro Vendramin; Michiel Vermeulen. Also, I have benefited from the posts to mathoverflow by Richard Borcherds, Robin Chapman, Steve Dalton, Leonid Positselski, Noah Snyder, Richard Stanley, Qiaochu Yuan, and others. A reference monnnn means question nnnn on mathoverflow.net and sxnnnn similarly refers to math.stackexchange.com. 1 A family should be distinguished from a set. For example, if f is the function Z Z=3Z sending an ! integer to its equivalence class, then f .i/ i Z is a set with three elements whereas .f .i//i Z is family with an infinite index set. f j 2 g 2 5 The theory of groups of finite order may be said to date from the time of Cauchy. To him are due the first attempts at classification with a view to forming a theory froma number of isolated facts. Galois introduced into the theory the exceedingly important idea of a [normal] sub-group, and the corresponding division of groups into simple and composite. Moreover, by shewing that to every equation of finite degree there corresponds a group of finite order on which all the properties of the equation depend, Galois indicated how far reaching the applications of the theory might be, and thereby contributed greatly, if indirectly, to its subsequent developement. Many additions were made, mainly by French mathematicians, during the middle part of the [nineteenth] century. The first connected exposition of the theory was given in the third edition of M. Serret’s “Cours d’Algebre` Superieure,´ ” which was published in 1866. This was followed in 1870 by M. Jordan’s “Traite´ des substitutions et des equations´ algebriques.´ ” The greater part of M. Jordan’s treatise is devoted to a developement of the ideas of Galois and to their application to the theory of equations. No considerable progress in the theory, as apart from its applications, was made till the appearance in 1872 of Herr Sylow’s memoir “Theor´ emes` sur les groupes de substitutions” in the fifth volume of the Mathematische Annalen. Since the date of this memoir, but more especially in recent years, the theory has