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Group Theory Group Theory J.S. Milne S3 Â1 2 3à r D 2 3 1 Â1 2 3à f D 1 3 2 Version 3.10 September 24, 2010 The first version of these notes was written for a first-year graduate algebra course. As in 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 (v3.10)}, year={2010}, note={Available at www.jmilne.org/math/}, pages={131} } 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. The multiplication table of S3 on the front page was produced by Group Explorer. Copyright c 1996, 2002, 2003, 2007, 2008, 2010 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Contents 3 1 Basic Definitions and Results 7 Definitions and examples . 7 Multiplication tables . 11 Subgroups . 12 Groups of small order . 14 Homomorphisms . 15 Cosets . 16 Normal subgroups . 18 Kernels and quotients . 20 Theorems concerning homomorphisms . 21 Direct products . 23 Commutative groups . 25 The order of ab ................................... 29 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 . 40 3 Automorphisms and Extensions 41 Automorphisms of groups . 41 Characteristic subgroups . 43 Semidirect products . 44 Extensions of groups . 48 The Holder¨ program. 50 Exercises . 51 4 Groups Acting on Sets 53 Definition and examples . 53 Permutation groups . 60 The Todd-Coxeter algorithm. 66 Primitive actions. 68 Exercises . 69 5 The Sylow Theorems; Applications 73 3 The Sylow theorems . 73 Alternative approach to the Sylow theorems . 77 Examples . 77 Exercises . 80 6 Subnormal Series; Solvable and Nilpotent Groups 81 Subnormal Series. 81 Solvable groups . 83 Nilpotent groups . 87 Groups with operators . 90 Krull-Schmidt theorem . 92 Exercises . 93 7 Representations of Finite Groups 95 Matrix representations . 95 Roots of 1 in fields . 96 Linear representations . 96 Maschke’s theorem . 97 The group algebra; semisimplicity . 99 Semisimple modules . 100 Simple F -algebras and their modules . 101 Semisimple F -algebras and their modules . 105 The representations of G ..............................107 The characters of G .................................108 The character table of a group . 111 Examples . 111 Exercises . 111 A Additional Exercises 113 B Solutions to the Exercises 117 C Two-Hour Examination 125 Bibliography 127 Index 129 4 NOTATIONS. We use the standard (Bourbaki) notations: N 0;1;2;::: ; Z is the ring of integers; Q D f g 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. 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 ; j j j j of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed 1 by 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 the- ory. 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 ver- sions 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 pro- grams. 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 comments for earlier versions of these notes: Dustin Clausen, Benoˆıt Claudon, Keith Conrad, Demetres Christofides, Adam Glesser, Sylvan Jacques, Martin Klazar, Mark Meckes, Victor Petrov, Efthymios Sofos, Dave Simp- son, Robert Thompson, 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 mo9990 means http://mathoverflow.net/questions/9990/). 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 from a 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 devel- opement 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 advanced 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 composees´ as the most important dichotomy in the theory of permutation groups. Moreover, in the Traite´, Jordan began building a database of finite simple groups — the alternating groups of degree at least 5 and most of the classical pro- jective 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 some- one 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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