J.S. Milne

1 group MilneS3rD 1 2 32 3 1 fD 1 2 31 3 2 Version 24, 2010 The first version of these notes was written for a first-year graduate algebra course. As inmost such courses, the notes concentrated on abstract groups and, in particular, on finitegroups. 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 groupsthemselves that are of interest, but also their linear representations. It is my intention (oneday) to expand the notes to take account of this, and to produce a volume that, while stillmodest in size (c200 pages), will provide a more comprehensive introduction to grouptheory for beginning graduate students in mathematics, physics, and related information@misc{milneGT,author={ Milne , James S.}}

2 },title={ group theory ( )},year={2010},note={Available at },pages={131}}Please send comments and corrections to me at the address on my (August 21, 1996). First version on the web; 57 (August 29,2003). Fixed many minor errors; numbering unchanged; 85 (September 1, 2007). Revised and expanded; 121 (May 17, 2008). Minor fixes and changes; 124 (September 21, 2009). Minor fixes; changed TeX styles; 127 (September 24, 2010). Many minor improvements; 131 multiplication table ofS3on the front page was produced by group 1996, 2002, 2003, 2007, 2008, 2010 paper copies for noncommercial personal use may be made without explicit permis-sion from the copyright Basic Definitions and Results7 Definitions and examples ..7 Multiplication tables.

3 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 ofab..29 Exercises ..292 Free Groups and Presentations; Coxeter Groups31 Free monoids ..31 Free groups ..32 Generators and relations ..35 Finitely presented groups ..37 Coxeter groups ..38 Exercises ..403 Automorphisms and Extensions41 Automorphisms of groups ..41 Characteristic subgroups ..43 Semidirect products ..44 Extensions of groups ..48 The H older program..50 Exercises ..514 Groups Acting on Sets53 Definition and examples ..53 Permutation groups ..60 The Todd-Coxeter.

4 66 Primitive actions..68 Exercises ..695 The Sylow Theorems; Applications733 The Sylow theorems ..73 Alternative approach to the Sylow theorems ..77 Examples ..77 Exercises ..806 Subnormal Series; Solvable and Nilpotent Groups81 Subnormal Series..81 Solvable groups ..83 Nilpotent groups ..87 Groups with operators ..90 Krull-Schmidt theorem ..92 Exercises ..937 Representations of Finite Groups95 Matrix representations ..95 Roots of1in fields ..96 Linear representations ..96 Maschke s theorem ..97 The group algebra; semisimplicity ..99 Semisimple modules .. 100 SimpleF-algebras and their modules .. 101 SemisimpleF-algebras and their modules .. 105 The representations ofG.. 107 The characters ofG.. 108 The character table of a group .

5 111 Examples .. 111 Exercises .. 111A Additional Exercises113B Solutions to the Exercises117C Two-Hour use the standard (Bourbaki) notations:NDf0;1;2;:::g;Zis the ring of integers;Qis the field of rational numbers;Ris the field of real numbers;Cis the field of complexnumbers;Fqis a finite field withqelements whereqis a power of a prime number. Inparticular,FpDZ=pZforpa prime integersmandn,mjnmeans thatmdividesn, ,n2mZ. Throughout the notes,pis a prime number, ,pD2;3;5;7;11;:::;1000000007;:::.Given an equivalence relation, denotes the equivalence class containing . Theempty set is denoted by;. The cardinality of a setSis denoted byjSj(sojSjis the numberof elements inSwhenSis finite). LetIandAbe sets; a family of elements ofAindexedbyI, , is a functioni7!

6 AiWI! are required to have an identity element1, and homomorphisms of rings arerequired to take1to1. An elementaof a ring is a unit if it has an inverse (elementbsuchthatabD1 Dba). The identity element of a ring is required to act as1on a module overthe Y Xis a subset ofY(not necessarily proper);XdefDY Xis defined to beY, or equalsYby definition;X Y Xis isomorphic toY;X'Y XandYare canonically isomorphic (or there is a given or unique isomorphism);PREREQUISITESAn undergraduate abstract algebra ALGEBRA PROGRAMSGAP is an open source computer algebra program, emphasizing computational group the-ory. To get started with GAP, I recommend going to Alexander Hulpke s ~hulpke/ you will find ver-sions of GAP for both Windows and Macs and a guide Abstract Algebra in GAP.

7 TheSage a front end for GAP and other pro-grams. I also recommend N. Carter s group Explorer exploring the structure of groups of small order. Earlier versions of these notes( ) described how to use Maple for computations in group thank the following for providing corrections and comments for earlier versions of thesenotes: Dustin Clausen, Beno t Claudon, Keith Conrad, Demetres Christofides, Adam Glesser,Sylvan Jacques, Martin Klazar, Mark Meckes, Victor Petrov, Efthymios Sofos, Dave Simp-son, Robert Thompson, Michiel , I have benefited from the posts to mathoverflow by Richard Borcherds, RobinChapman, Steve Dalton, Leonid Positselski, Noah Snyder, Richard Stanley, Qiaochu Yuan,and others (a reference mo9990 ).1A family should be distinguished from a set.

8 For example, iffis the functionZ!Z=3 Zsending aninteger to its equivalence class, a set with three elements familywith an infinite index theory of groups of finite order may be said to date from the time of Cauchy. Tohim are due the first attempts at classification with a view to forming a theory from anumber of isolated facts. Galois introduced into the theory the exceedingly importantidea of a [normal] sub- group , and the corresponding division of groups into simpleand composite. Moreover, by shewing that to every equation of finite degree therecorresponds 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 therebycontributed greatly, if indirectly, to its subsequent additions were made, mainly by French mathematicians, during the middlepart of the [nineteenth] century.

9 The first connected exposition of the theory was givenin the third edition of M. Serret s Cours d Alg`ebre Sup erieure, which was publishedin 1866. This was followed in 1870 by M. Jordan s Trait e des substitutions et des equations alg ebriques. 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 considerable progress in the theory , as apart from its applications, was madetill the appearance in 1872 of Herr Sylow s memoir Th eor`emes sur les groupes desubstitutions in the fifth volume of theMathematische the date of thismemoir, but more especially in recent years, the theory has advanced Burnside, theory of Groups of Finite Order, introduced the concept of a normal subgroup in 1832, and Camille Jordan in thepreface to hisTrait e.

10 In 1870 flagged Galois distinction between groupes simplesand groupes compos ees as the most important dichotomy in the theory of permutationgroups. Moreover, in theTrait e, Jordan began building a database of finite simplegroups the alternating groups of degree at least5and most of the classical pro-jective linear groups over fields of prime cardinality. Finally, in 1872, Ludwig Sylowpublished his famous theorems on subgroups of prime power Solomon, Bull. Amer. Math. Soc., 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 leastwith the current methods of classification via centralizers of involutions, is that everysimple group has to be tested to see if it leads to new simple groups containing it inthe centralizer of an involution.