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Group Theory Group Theory J.S. Milne Version 3.11 March 28, 2011 S3 Â1 2 3à r D 2 3 1 Â1 2 3à f D 1 3 2 The first version of these notes was written for a first-year grad- uate algebra course. As in most such courses, the notes concen- trated 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 mathemat- ics, physics, and related fields. BibTeX information @misc{milneGT, author={Milne, James S.}, title={Group Theory (v3.11)}, year={2011}, note={Available at www.jmilne.org/math/}, pages={135} } Please send comments and corrections to me at the address on my website www.jmilne.org/math/. v2.01 (August 21, 1996). First version on the web; 57 pages. v2.11 (August 29, 2003). Fixed many minor errors; numbering unchanged; 85 pages. v3.00 (September 1, 2007). Revised and expanded; 121 pages. v3.01 (May 17, 2008). Minor fixes and changes; 124 pages. v3.02 (September 21, 2009). Minor fixes; changed TeX styles; 127 pages. v3.10 (September 24, 2010). Many minor improvements; 131 pages. v3.11 (March 28, 2011). Minor additions; 135 pages. The multiplication table of S3 on the front page was produced by Group Explorer. Copyright c 1996, 2002, 2003, 2007, 2008, 2010, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Contents Contents 3 1 Basic Definitions and Results 11 Definitions and examples . 11 Multiplication tables . 21 Subgroups . 22 Groups of small order . 28 Homomorphisms . 30 Cosets . 32 Normal subgroups . 35 Kernels and quotients . 39 Theorems concerning homomorphisms . 42 Direct products . 46 Commutative groups . 49 The order of ab .................... 57 Exercises . 58 2 Free Groups and Presentations; Coxeter Groups 61 Free monoids . 62 Free groups . 63 3 Generators and relations . 69 Finitely presented groups . 74 Coxeter groups . 76 Exercises . 80 3 Automorphisms and Extensions 83 Automorphisms of groups . 83 Characteristic subgroups . 88 Semidirect products . 89 Extensions of groups . 97 The Holder¨ program. 101 Exercises . 105 4 Groups Acting on Sets 109 Definition and examples . 109 Permutation groups . 124 The Todd-Coxeter algorithm. 136 Primitive actions. 139 Exercises . 142 5 The Sylow Theorems; Applications 147 The Sylow theorems . 148 Alternative approach to the Sylow theorems . 155 Examples . 156 Exercises . 163 6 Solvable and Nilpotent Groups 165 Subnormal Series. 165 Solvable groups . 170 Nilpotent groups . 176 4 Groups with operators . 183 Krull-Schmidt theorem . 187 Exercises . 189 7 Representations of Finite Groups 191 Matrix representations . 192 Roots of 1 in fields . 193 Linear representations . 194 Maschke’s theorem . 195 The group algebra; semisimplicity . 200 Semisimple modules . 201 Simple F -algebras and their modules . 203 Semisimple F -algebras and their modules . 212 The representations of G . 216 The characters of G . 219 The character table of a group . 225 Examples . 225 Exercises . 225 A Additional Exercises 229 B Solutions to the Exercises 239 C Two-Hour Examination 253 Bibliography 259 Index 263 5 NOTATIONS. We use the standard (Bourbaki) notations: N 0;1;2;::: ; Z D f g is the ring of integers; Q is the 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. For integers m and n,Dm n means that m divides n, i.e., j n mZ. Throughout the notes, p is a prime number, i.e., p 2 2;3;5;7;11;:::;1000000007;:::. DGiven an equivalence relation, Œ denotes the equivalence class containing . The empty set is denoted by . The car- dinality of a set S is denoted by S (so S is the; number of elements in S when S is finite). Letj jI andj Aj be sets; a family of elements of A indexed by I , denoted .ai /i I , is a function 1 2 i ai I A. 7!RingsW are! required to have an identity element 1, and ho- momorphisms 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 1Don aD module over the ring. X YX is a subset of Y (not necessarily proper); def X YX is defined to be Y , or equals Y by definition; X D YX is isomorphic to Y ; X YX and Y are canonically isomorphic (or there is a given or unique isomorphism); ' 1A family should be distinguished from a set. For example, if f is the func- tion Z Z=3Z sending an integer to its equivalence class, then f .i/ i Z ! f j 2 g is a set with three elements whereas .f .i//i Z is family with an infinite index set. 2 6 PREREQUISITES An undergraduate “abstract algebra” course. COMPUTER ALGEBRA PROGRAMS GAP is an open source computer algebra program, emphasiz- ing computational group theory. To get started with GAP, I recommend going to Alexander Hulpke’s page http://www. math.colostate.edu/~hulpke/CGT/education.html where you will find versions of GAP for both Windows and Macs and a guide “Abstract Algebra in GAP”. The Sage page http://www.sagemath.org/ provides a front end for GAP and other programs. I also recommend N. Carter’s “Group Explorer” http://groupexplorer.sourceforge.net 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 com- ments for earlier versions of these notes: Tony Bruguier, Dustin Clausen, Benoˆıt Claudon, Keith Conrad, Demetres Christofides, Adam Glesser, Sylvan Jacques, Martin Klazar, Mark Meckes, Victor Petrov, Efthymios Sofos, Dave Simpson, Robert Thompson, Michiel Vermeulen. Also, I have benefited from the posts to mathover- flow by Richard Borcherds, Robin Chapman, Steve Dal- ton, Leonid Positselski, Noah Snyder, Richard Stanley, 7 Qiaochu Yuan, and others (a reference monnnn means http://mathoverflow.net/questions/nnnn/). 8 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 classifica- tion with a view to forming a theory from a number of isolated facts. Galois introduced into the theory the exceedingly impor- tant idea of a [normal] sub-group, and the corresponding divi- sion 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 de- pend, 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 mathemati- cians, 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 ap- plications, 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 ad- vanced continuously. W. Burnside, Theory of Groups of Finite Order, 1897. Galois introduced the concept of a normal subgroup in 1832, and Camille Jordan in the preface to his Traite.´ . in 1870 flagged Galois’ distinction between groupes simples and groupes com- posees´ as the most important dichotomy in the theory of permu- tation groups. Moreover, in the Traite´, Jordan began building a database of finite simple groups — the alternating groups of de- gree at least 5 and most of the classical projective linear groups over fields of prime cardinality. Finally, in 1872, Ludwig Sylow published his famous theorems on subgroups of prime power order. R. Solomon, Bull. Amer. Math. Soc., 2001. Why are the finite simple groups classifiable? It is unlikely that there is any easy reason why a classification is possible, unless someone comes up with a completely new way to classify groups. One problem, at least with the current methods of classification via centralizers of involutions, is that every simple group has to be tested to see if it leads to new simple groups containing it in the centralizer of an involution. For example, when the baby monster was discovered, it had a double cover, which was a potential centralizer of an involution in a larger simple group, which turned out to be the monster. The monster happens to have no double cover so the process stopped there, but without checking every finite simple group there seems no obvious reason why one cannot have an infinite chain of larger and larger sporadic groups, each of which has a double cover that is a centralizer of an involution in the next one.
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