Lecture notes Group theory Physics

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Group Theory J.S. Milne S 3   1 2 3 r D 2 3 1   1 2 3 f D 1 3 2 Version 3.14 March 17, 2017The 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 miscmilneGT, author=Milne, James S., title=Group Theory (v3.13), year=2013, note=Available at www.jmilne.org/math/, pages=127 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. v3.12 (April 9, 2012). Minor fixes; 133 pages. v3.13 (March 15, 2013). Minor fixes; 133 pages. v3.14 (March 17, 2017). Minor fixes; 135 pages. The multiplication table ofS on the front page was produced by Group Explorer. 3 Copyright c 1996–2017 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 7 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 ofab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 Automorphisms and Extensions 43 Automorphisms of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Characteristic subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Semidirect products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Extensions of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 The Holder ¨ program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 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 3The 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 of1 in fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Linear representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Maschke’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 The group algebra; semisimplicity . . . . . . . . . . . . . . . . . . . . . . . . . 103 Semisimple modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 SimpleF -algebras and their modules . . . . . . . . . . . . . . . . . . . . . . . . 105 SemisimpleF -algebras and their modules . . . . . . . . . . . . . . . . . . . . . 109 The representations ofG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 The characters ofG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 The character table of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 A Additional Exercises 117 B Solutions to the Exercises 121 C Two-Hour Examination 129 Bibliography 131 Index 133 4NOTATIONS. We use the standard (Bourbaki) notations: NDf0;1;2;:::g;Z 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;F is a finite field withq elements whereq is a power of a prime number. In q particular,F DZ=pZ forp a prime number. p For integersm andn,mjn means thatm dividesn, i.e.,n2mZ. Throughout the notes, p is a prime number, i.e.,pD2;3;5;7;11;:::;1000000007;:::. Given an equivalence relation,Œ denotes the equivalence class containing. The empty set is denoted by;. The cardinality of a setS is denoted byjSj (sojSj is the number of elements inS whenS is finite). LetI andA be sets; a family of elements ofA indexed by 1 I , denoted.a / , is a functioni7aWIA. i i2I i Rings are required to have an identity element1, and homomorphisms of rings are required to take1 to1. An elementa of a ring is a unit if it has an inverse (elementb such thatabD1Dba). The identity element of a ring is required to act as1 on a module over the ring. XY X is a subset ofY (not necessarily proper); def XDY X is defined to beY , or equalsY by definition; XY X is isomorphic toY ; X'Y X andY 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; Tony Bruguier; Dustin Clausen; Benoˆ ıt Claudon; Keith Conrad; Demetres Christofides; Adam Glesser; Darij Grinberg; Sylvan Jacques; Martin Klazar; Mark Meckes; Victor Petrov; Diego Silvera; Efthymios Sofos; Dave Simpson; David Speyer; 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 meanshttp://mathoverflow.net/questions/nnnn/ and a reference sxnnnn meanshttp://math.stackexchange.com/questions/nnnn/). 1 A family should be distinguished from a set. For example, iff is the functionZZ=3Z sending an integer to its equivalence class, thenff.i/ji2Zg is a set with three elements whereas.f.i// is family with i2Z an infinite index set. 5The 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’Algebr ` e Superieur ´ e,” 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 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 least5 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. Because of this problem (among others), it was unclear until quite late in the classification whether there would be a finite or infinite number of sporadic groups. Richard Borcherds, mo38161.CHAPTER 1 Basic Definitions and Results The axioms for a group are short and natural. . . . Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on nu- merous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists. Richard Borcherds, in Mathematicians 2009. Group theory is the study of symmetries. Definitions and examples DEFINITION 1.1 A group is a setG together with a binary operation .a;b/7abWGGG satisfying the following conditions: G1: (associativity) for alla;b;c2G, .ab/cDa.bc/I G2: (existence of a neutral element) there exists an elemente2G such that aeDaDea (1) for alla2G; 0 G3: (existence of inverses) for eacha2G, there exists ana2G such that 0 0 aaDeDaa: We usually abbreviate.G;/ toG. Also, we usually writeab forab and1 fore; al- ternatively, we writeaCb forab and 0 fore. In the first case, the group is said to be multiplicative, and in the second, it is said to be additive. 1.2 In the following,a;b;::: are elements of a groupG. 78 1. BASIC DEFINITIONS AND RESULTS 0 (a) An elemente satisfying (1) is called a neutral element. Ife is a second such element, 0 0 thene Dee De. In fact,e is the unique element ofG such thateeDe (apply G3). (b) IfbaDe andacDe, then bDbeDb.ac/D.ba/cDecDc: 0 Hence the elementa in (G3) is uniquely determined bya. We call it the inverse ofa, 1 and denote ita (or the negative ofa, and denote ita). (c) Note that (G1) shows that the product of any ordered triplea ,a ,a of elements of 1 2 3 G is unambiguously defined: whether we forma a first and then.a a /a , ora a 1 2 1 2 3 2 3 first and thena .a a /, the result is the same. In fact, (G1) implies that the product of 1 2 3 any orderedn-tuplea ,a ,. . . ,a of elements ofG is unambiguously defined. We 1 2 n prove this by induction onn. In one multiplication, we might end up with .a a /.a a / (2) 1 i iC1 n as the final product, whereas in another we might end up with .a a /.a a /: (3) 1 j jC1 n Note that the expression within each pair of parentheses is well defined because of the induction hypotheses. Thus, ifiDj , (2) equals (3). Ifi¤j , we may supposeij . Then  .a a /.a a /D.a a / .a a /.a a / 1 i iC1 n 1 i iC1 j jC1 n  .a a /.a a /D .a a /.a a / .a a / 1 j jC1 n 1 i iC1 j jC1 n and the expressions on the right are equal because of (G1). 1 1 1 (d) The inverse of a a a is a a a , i.e., the inverse of a product is the 1 2 n n n1 1 product of the inverses in the reverse order. (e) (G3) implies that the cancellation laws hold in groups, abDacH) bDc; baDcaH) bDc 1 (multiply on left or right bya ). Conversely, ifG is finite, then the cancellation laws imply (G3): the mapx7axWGG is injective, and hence (by counting) bijective; in particular,e is in the image, and soa has a right inverse; similarly, it has a left inverse, and the argument in (b) above shows that the two inverses are equal. 0 0 Two groups.G;/ and.G;/ are isomorphic if there exists a one-to-one correspon- 0 0 0 0 0 0 denceaa ,GG , such that.ab/Da b for alla;b2G. The orderjGj of a groupG is its cardinality. A finite group whose order is a power of a primep is called ap-group. For an elementa of a groupG, define 8 aaa n0 .n copies ofa/ n a D e nD0 : 1 1 1 1 a a a n0 (jnj copies ofa )Definitions and examples 9 The usual rules hold: m n mCn m n mn a a Da ; .a / Da ; allm;n2Z: (4) It follows from (4) that the set n fn2Zja Deg n is an ideal inZ, and so equalsmZ for some integerm0. WhenmD0,a ¤e unless nD0, anda is said to have infinite order. Whenm¤0, it is the smallest integerm0 m 1 m1 such thata De, anda is said to have finite orderm. In this case,a Da , and n a De” mjn: EXAMPLES 1.3 LetC be the group.Z;C/, and, for an integerm1, letC be the group.Z=mZ;C/. 1 m 1.4 Permutation groups. LetS be a set and let Sym.S/ be the set of bijections WSS. We define the product of two elements of Sym.S/ to be their composite: D  : In other words,. /.s/D . .s// for alls2S. For any ; ; 2 Sym.S/ ands2S, ..  / /.s/D.  /. .s//D . . .s///D. .  //.s/; (5) and so associativity holds. The identity maps7s is an identity element for Sym.S/, and inverses exist because we required the elements of Sym.S/ to be bijections. Therefore Sym.S/ is a group, called the group of symmetries ofS. For example, the permutation group onn lettersS is defined to be the group of symmetries of the setf1;:::;ng — it has n ordernŠ. 1.5 WhenG andH are groups, we can construct a new groupGH , called the (direct) product ofG andH . As a set, it is the cartesian product ofG andH , and multiplication is defined by 0 0 0 0 .g;h/.g;h/D.gg;hh/: 1 1.6 A groupG is commutative (or abelian) if abDba; alla;b2G: In a commutative group, the product of any finite (not necessarily ordered) familyS of elements is well defined, for example, the empty product ise. Usually, we write commutative groups additively. With this notation, Equation (4) becomes: maCnaD.mCn/a; m.na/Dmna: WhenG is commutative, m.aCb/DmaCmb form2Z anda;b2G, 1 “Abelian group” is more common than “commutative group”, but I prefer to use descriptive names where possible.10 1. BASIC DEFINITIONS AND RESULTS and so the map .m;a/7maWZGG makesA into aZ-module. In a commutative groupG, the elements of finite order form a subgroupG ofG, called the torsion subgroup. tors 1.7 LetF be a field. Thenn matrices with coefficients inF and nonzero determinant form a group GL .F/ called the general linear group of degreen. For a finite dimensional n F -vector spaceV , theF -linear automorphisms ofV form a group GL.V/ called the general linear group ofV . Note that ifV has dimensionn, then the choice of a basis determines an isomorphism GL.V/ GL .F/ sending an automorphism to its matrix with respect to the n basis. 1.8 LetV be a finite dimensional vector space over a fieldF . A bilinear form onV is a mappingWVVF that is linear in each variable. An automorphism of such a is an isomorphism WVV such that . v; w/D.v;w/ for allv;w2V: (6) The automorphisms of form a group Aut./. Lete ;:::;e be a basis forV , and let 1 n PD..e ;e // i j 1i;jn be the matrix of. The choice of the basis identifies Aut./ with the group of invertible 2 matricesA such that T A PADP . (7) When is symmetric, i.e., .v;w/D.w;v/ allv;w2V; and nondegenerate, Aut./ is called the orthogonal group of. When is skew-symmetric, i.e., .v;w/D.w;v/ allv;w2V; and nondegenerate, Aut./ is called the symplectic group of. In this case, there exists a basis forV for which the matrix of is   0 I m J D ; 2mDn; 2m I 0 m 2 n When we use the basis to identifyV withF , the pairing becomes 0 1 0 1 a b b 1 1 1 : : A : A ; 7.a ;:::;a /P : : : 1 n : : : : a n b b n n a a 1 1 : : IfA is the matrix of with respect to the basis, then corresponds to the map 7A :Therefore, : : : : a a n n (6) becomes the statement that 0 1 0 1 0 1 b b a b 1 1 1 1 T n : : : : A A A .a ;:::;a /A PA D.a ;:::;a /P for all ; 2F : 1 n : 1 n : : : : : : : a n b b b n n n n On examining this statement on the standard basis vectors forF , we see that it is equivalent to (7).Multiplication tables 11 and the group of invertible matricesA such that T A J ADJ 2m 2m is called the symplectic group Sp . 2m REMARK 1.9 A setS together with a binary operation.a;b/7abWSSS is called a magma. When the binary operation is associative,.S;/ is called a semigroup. The product Q def ADa a 1 n of any sequenceAD.a / of elements in a semigroupS is well-defined (see 1.2(c)), i 1in and for any pairA andB of such sequences, Q Q Q . A/. B/D .AtB/. (8) Let; be the empty sequence, i.e., the sequence of elements inS indexed by the empty set. Q What should ; be? Clearly, we should have Q Q Q Q Q Q Q . ;/. A/D .;tA/D AD .At;/D. A/. ;/: Q In other words, ; should be a neutral element. A semigroup with a neutral element is called a monoid. In a monoid, the product of any finite (possibly empty) sequence of elements is well-defined, and (8) holds. ASIDE 1.10 (a) The group conditions (G2,G3) can be replaced by the following weaker conditions 0 (existence of a left neutral element and left inverses): (G2 ) there exists ane such thateaDa for 0 0 0 alla; (G3 ) for eacha2G, there exists ana2G such thataaDe. To see that these imply (G2) 0 0 00 0 00 0 and (G3), leta2G, and apply (G3 ) to finda anda such thataaDe anda aDe. Then  0 0 00 0 0 00 0 0 00 0 aaDe.aa/D.a a/.aa/Da  .aa/a Da aDe; whence (G3), and 0 0 aDeaD.aa/aDa.aa/Dae; whence (G2). (b) A group can be defined to be a setG with a binary operation satisfying the following 0 conditions: (g1) is associative; (g2)G is nonempty; (g3) for eacha2G, there exists ana2G 0 such thataa is neutral. As there is at most one neutral element in a set with an associative binary operation, these conditions obviously imply those in (a). They are minimal in the sense that there exist sets with a binary operation satisfying any two of them but not the third. For example,.N;C/ satisfies (g1) and (g2) but not (g3); the empty set satisfies (g1) and (g3) but not (g2); the set of22 matrices with coefficents in a field and withABDABBA satisfies (g2) and (g3) but not (g1). Multiplication tables A binary operation on a finite set can be described by its multiplication table: e a b c ::: e ee ea eb ec ::: 2 a ae a ab ac ::: 2 b be ba b bc ::: 2 c ce ca cb c ::: : : : : : : : : : : : : : : :12 1. BASIC DEFINITIONS AND RESULTS The elemente is an identity element if and only if the first row and column of the table simply repeat the elements. Inverses exist if and only if each element occurs exactly once in each row and in each column (see 1.2e). If there aren elements, then verifying the 3 associativity law requires checkingn equalities. For the multiplication table ofS , see the front page. Note that each colour occurs 3 exactly once in each row and and each column. This suggests an algorithm for finding all groups of a given finite ordern, namely, list all possible multiplication tables and check the axioms. Except for very smalln, this is not 2 practical The table hasn positions, and if we allow each position to hold any of then 2 n elements, then that gives a total ofn possible tables very few of which define groups. For 64 example, there are8 D6277101735386680763835789423207666416102355444464 034512896 binary operations on a set with8 elements, but only five isomorphism classes of groups of order8 (see 4.21). Subgroups PROPOSITION 1.11 LetS be a nonempty subset of a groupG. If S1: a;b2S H) ab2S, and 1 S2: a2S H) a 2S; then the binary operation onG makesS into a group. PROOF. (S1) implies that the binary operation onG defines a binary operationSSS onS, which is automatically associative. By assumptionS contains at least one elementa, 1 1 its inversea , and the producteDaa . Finally (S2) shows that the inverses of elements inS lie inS. 2 A nonempty subsetS satisfying (S1) and (S2) is called a subgroup ofG. WhenS is 2 finite, condition (S1) implies (S2): leta2S; thenfa;a ;:::gS, and soa has finite order, n 1 n1 saya De; nowa Da 2S. The example.N;C/.Z;C/ shows that (S1) does not imply (S2) whenS is infinite. EXAMPLE 1.12 The centre of a groupG is the subset Z.G/Dfg2GjgxDxg for allx2Gg: It is a subgroup ofG. PROPOSITION 1.13 An intersection of subgroups ofG is a subgroup ofG: PROOF. It is nonempty because it containse, and (S1) and (S2) obviously hold. 2 REMARK 1.14 It is generally true that an intersection of subobjects of an algebraic object is a subobject. For example, an intersection of subrings of a ring is a subring, an intersection of submodules of a module is a submodule, and so on. PROPOSITION 1.15 For any subsetX of a groupG, there is a smallest subgroup ofG containingX. It consists of all finite products of elements ofX and their inverses (repetitions allowed).Subgroups 13 PROOF. The intersectionS of all subgroups ofG containingX is again a subgroup con- tainingX, and it is evidently the smallest such group. ClearlyS contains withX, all finite products of elements ofX and their inverses. But the set of such products satisfies (S1) and (S2) and hence is a subgroup containingX. It therefore equalsS. 2 The subgroupS given by the proposition is denotedhXi, and is called the subgroup generated byX. For example,h;iDfeg. If every element ofX has finite order, for example, ifG is finite, then the set of all finite products of elements ofX is already a group and so equalshXi. We say thatX generatesG ifGDhXi, i.e., if every element ofG can be written as a finite product of elements fromX and their inverses. Note that the order of an elementa of a group is the order of the subgrouphai it generates. EXAMPLES 1.16 The cyclic groups. A group is said to be cyclic if it is generated by a single element, i.e., ifGDhri for somer2G. Ifr has finite ordern, then 2 n1 i GDfe;r;r ;:::;r gC ; r i modn; n andG can be thought of as the group of rotational symmetries about the centre of a regular polygon withn-sides. Ifr has infinite order, then i 1 i i GDf:::;r ;:::;r ;e;r;:::;r ;:::gC ; r i: 1 Thus, up to isomorphism, there is exactly one cyclic group of ordern for eachn1. In future, we shall loosely useC to denote any cyclic group of ordern (not necessarilyZ=nZ n orZ). 3 1.17 The dihedral groupsD . Forn3,D is the group of symmetries of a regular n n 4 polygon withn-sides. Number the vertices1;:::;n in the counterclockwise direction. Letr be the rotation through2=n about the centre of polygon (soi7iC1 modn/, and lets be the reflection in the line (= rotation about the line) through the vertex1 and the centre of the polygon (soi7nC2i modn). For example, the pictures s s 1 1 r 8  r 11 11 sD sD 24 23 :  2 4  33 rD1231 rD12341 2 3 3 3 This group is denotedD orD depending on whether the author is viewing it abstractly or concretely as 2n n the symmetries of ann-polygon (or perhaps on whether the author is a group theorist or not; see mo48434). 4 More formally, D can be defined to be the subgroup ofS generated byrWi7iC1 (modn/ and n n sWi7nC2i (modn). Then all the statements concerningD can proved without appealing to geometry. n14 1. BASIC DEFINITIONS AND RESULTS illustrate the groupsD andD . In the general case 3 4 n 2 1 n1 r DeI s DeI srsDr (sosrDr s/: These equalites imply that n1 n1 D Dfe;r;:::;r ;s;rs;:::;r sg; n and it is clear from the geometry that the elements of the set are distinct, and sojDjD2n. n Lett be the reflection in the line through the midpoint of the side joining the vertices1 and2 and the centre of the polygon (soi7nC3i modn/. ThenrDts, because s t i7nC2i7nC3.nC2i/DiC1 modn: HenceD Dhs;ti and n 2 2 n n s De; t De; .ts/ DeD.st/ : We defineD to beC Df1;rg andD to beC C Df1;r;s;rsg. The groupD 1 2 2 2 2 2 is also called the Klein Viergruppe or, more simply, the 4-group. Note thatD is the full 3 group of permutations off1;2;3g. It is the smallest noncommutative group. By adding a tick at each vertex of a regular polygon, we can reduce its symmetry group fromD toC . By adding a line from the centre of the polygon to the vertex1, we reduce n n its symmetry group tohsi. Physicist like to say that we have “broken the symmetry”.   p  0 1 01 p 1.18 The quaternion groupQ: LetaD andbD . Then 10 1 0 4 2 2 1 3 3 a De; a Db ; bab Da (sobaDa b). The subgroup of GL .C/ generated bya andb is 2 2 3 2 3 QDfe;a;a ;a ;b;ab;a b;a bg: The groupQ can also be described as the subsetf1;i;j;kg of the quaternion algebra H. Recall that HDR1RiRjRk with the multiplication determined by 2 2 i D1Dj ; ijDkDji: The mapi7a,j7b extends uniquely to a homomorphismHM .C/ ofR-algebras, 2 which maps the grouphi;ji isomorphically ontoha;bi. 1.19 Recall thatS is the permutation group onf1;2;:::;ng. A transposition is a permu- n tation that interchanges two elements and leaves all other elements unchanged. It is not difficult to see thatS is generated by transpositions (see (4.26) below for a more precise n statement).Groups of small order 15 Groups of small order FornD6, there are three groups, a groupC , 6 and two groupsC C andS . 2 3 3 Cayley, American J. Math. 1 (1878), p. 51. For each primep, there is only one group of orderp, namelyC (see 1.28 below). In the p following table,cCnDt means that there arec commutative groups andn noncommutative groups (up to isomorphism, of course). jGj cCnDt Groups Ref. 4 2C0D2 C ,C C 4.18 4 2 2 6 1C1D2 C ;S 4.23 6 3 8 3C2D5 C ,C C ,C C C ;Q,D 4.21 8 2 4 2 2 2 4 9 2C0D2 C ,C C 4.18 9 3 3 10 1C1D2 C ;D 5.14 10 5 12 2C3D5 C ,C C ;C S ,A ,C ÌC 5.16 12 2 6 2 3 4 4 3 14 1C1D2 C ;D 5.14 14 7 15 1C0D1 C 5.14 15 16 5C9D14 See Wild 2005 18 2C3D5 C ,C C ;D ;S C ,.C C /ÌC 18 3 6 9 3 3 3 3 2 5 2 2 20 2C3D5 C ,C C ;D ,C ÌC ,ha;bja Db Dc Dabci 20 2 10 10 5 4 3 7 2 21 1C1D2 C ;ha;bja Db D1,baDabi 21 22 1C1D2 C ;D 5.14 22 11 24 3C12D15 groupprops.subwiki.org/wiki/Groups of order 24 5 2 2 5 Hereha;bja Db Dc Dabci is the group with generatorsa andb and relationsa D 2 2 b Dc Dabc (see Chapter 2). It is the dicyclic group. Roughly speaking, the more high powers of primes dividen, the more groups of ordern there should be. In fact, iff.n/ is the number of isomorphism classes of groups of ordern, then 2 2 . Co.1//e.n/ 27 f.n/n wheree.n/ is the largest exponent of a prime dividingn ando.1/0 ase.n/1 (see Pyber 1993). By 2001, a complete irredundant list of groups of order2000 had been found — up to 5 isomorphism, there are exactly 49,910,529,484 (Besche et al. 2001). 5 In fact Besche et al. did not construct the groups of order 1024 individually, but it is known that there are 49487365422 groups of that order. The remaining 423164062 groups of order up to 2000 (of which 408641062 have order 1536) are available as libraries in GAP and Magma. I would guess that 2048 is the smallest number such that the exact number of groups of that order is unknown (Derek Holt, mo46855; Nov 21, 2010).16 1. BASIC DEFINITIONS AND RESULTS Homomorphisms 0 0 DEFINITION 1.20 A homomorphism from a groupG to a secondG is a map WGG such that .ab/D .a/ .b/ for alla;b2G. An isomorphism is a bijective homomorphism.  For example, the determinant map detWGL .F/F is a homomorphism. n 1.21 Let be a homomorphism. For any elementsa ;:::;a ofG, 1 m .a a /D .a .a a // 1 m 1 2 m D .a / .a a / 1 2 m  D .a / .a /, 1 m and so homomorphisms preserve all products. In particular, form1, m m .a /D .a/ : (9) Moreover .e/D .ee/D .e/ .e/, and so .e/De (apply 1.2a). Also 1 1 1 1 aa DeDa aH) .a/ .a /DeD .a / .a/; 1 1 and so .a /D .a/ . It follows that (9) holds for allm2Z, and so a homomorphism of commutative groups is also a homomorphism ofZ-modules. As we noted above, each row of the multiplication table of a group is a permutation of the elements of the group. As Cayley pointed out, this allows one to realize the group as a group of permutations. THEOREM 1.22 (CAYLEY) There is a canonical injective homomorphism WG Sym.G/: PROOF. Fora2G, definea WGG to be the mapx7ax (left multiplication bya). For L x2G, .a b /.x/Da .b .x//Da .bx/DabxD.ab/ .x/; L L L L L L and so.ab/ Da b . Ase D id, this implies that L L L L 1 1 a .a / D idD.a / a ; L L L L and soa is a bijection, i.e., a 2 Sym.G/. Hencea7a is a homomorphismG L L L Sym.G/, and it is injective because of the cancellation law. 2 COROLLARY 1.23 A finite group of ordern can be realized as a subgroup ofS . n PROOF. List the elements of the group asa ;:::;a . 1 n 2 Unfortunately, unlessn is small,S is too large to be manageable. We shall see later n (4.22) thatG can often be embedded in a permutation group of much smaller order thannŠ.Cosets 17 Cosets For a subsetS of a groupG and an elementa ofG, we let aSDfasjs2Sg SaDfsajs2Sg: Because of the associativity law,a.bS/D.ab/S , and so we can denote this set unambigu- ously byabS: WhenH is a subgroup ofG, the sets of the formaH are called the left cosets ofH inG, and the sets of the formHa are called the right cosets ofH inG. Becausee2H , aHDH if and only ifa2H . 2 EXAMPLE 1.24 LetGD.R ;C/, and letH be a subspace of dimension1 (line through the origin). Then the cosets (left or right) ofH are the linesaCH parallel toH . PROPOSITION 1.25 LetH be a subgroup of a groupG. (a) An elementa ofG lies in a left cosetC ofH if and only ifCDaH: (b) Two left cosets are either disjoint or equal. 1 (c)aHDbH if and only ifa b2H: (d) Any two left cosets have the same number of elements (possibly infinite). PROOF. (a) Certainlya2aH . Conversely, ifa lies in the left cosetbH , thenaDbh for someh, and so aHDbhHDbH: 0 (b) IfC andC are not disjoint, then they have a common elementa, andCDaH and 0 C DaH by (a). 1 1 1 (c) If a b2H , then HDa bH , and so aHDaa bHDbH . Conversely, if 1 1 aHDbH , thenHDa bH , and soa b2H . 1 (d) The map.ba / Wah7bh is a bijectionaHbH: L 2 6 The index.GWH/ ofH inG is defined to be the number of left cosets ofH inG. For example,.GW1/ is the order ofG. As the left cosets ofH inG coverG, (1.25b) shows that they form a partitionG. In other words, the condition “a andb lie in the same left coset” is an equivalence relation on G. THEOREM 1.26 (LAGRANGE) IfG is finite, then .GW1/D.GWH/.HW1/: In particular, the order of every subgroup of a finite group divides the order of the group. PROOF. The left cosets ofH inG form a partition ofG, there are.GWH/ of them, and each left coset has.HW1/ elements. 2 COROLLARY 1.27 The order of each element of a finite group divides the order of the group. 6 More formally,.GWH/ is the cardinality of the setfaHja2Gg.18 1. BASIC DEFINITIONS AND RESULTS PROOF. Apply Lagrange’s theorem toHDhgi, recalling that.HW1/D order.g/. 2 EXAMPLE 1.28 IfG has orderp, a prime, then every element ofG has order1 orp. But onlye has order1, and soG is generated by any elementa¤e. In particular,G is cyclic and soGC . This shows, for example, that, up to isomorphism, there is only one group p of order1;000;000;007 (because this number is prime). In fact there are only two groups of order1;000;000;014;000;000;049 (see 4.18). 1 1 1 1 1.29 For a subsetS ofG, letS Dfg jg2Sg. Then.aH/ is the right cosetHa , 1 1 1 and.Ha/ Da H . ThereforeS7S defines a one-to-one correspondence between 1 the set of left cosets and the set of right cosets under whichaHHa . Hence.GWH/ is also the number of right cosets ofH inG: But, in general, a left coset will not be a right coset (see 1.34 below). 1.30 Lagrange’s theorem has a partial converse: if a primep dividesmD.GW1/, thenG n has an element of orderp (Cauchy’s theorem 4.13); if a prime powerp dividesm, thenG n has a subgroup of orderp (Sylow’s theorem 5.2). However, note that the4-groupC C 2 2 has order4, but has no element of order4, andA has order12, but has no subgroup of order 4 6 (see Exercise 4-15). More generally, we have the following result. PROPOSITION 1.31 For any subgroupsHK ofG, .GWK/D.GWH/.HWK/ (meaning either both are infinite or both are finite and equal). F F PROOF. WriteGD g H (disjoint union), andHD h K (disjoint union). On i j i2I j2J F multiplying the second equality byg , we find thatg HD g h K (disjoint union), i i i j j2J F and soGD g h K (disjoint union). This shows that i j i;j2IJ .GWK/DjIjjJjD.GWH/.HWK/: 2 Normal subgroups WhenS andT are two subsets of a groupG, we let STDfstjs2S,t2Tg: Because of the associativity law,R.ST/D.RS/T , and so we can denote this set unambigu- ously asRST . 1 A subgroupN ofG is normal, denotedNGG, ifgNg DN for allg2G. 1 REMARK 1.32 To show thatN is normal, it suffices to check thatgNg N for allg, 1 because multiplying this inclusion on the left and right withg andg respectively gives 1 1 1 the inclusionNg Ng, and rewriting this withg forg gives thatNgNg for all g. However, the next example shows that there can exist a subgroupN of a groupG and an 1 1 elementg ofG such thatgNg N butgNg ¤N .Normal subgroups 19   1n EXAMPLE 1.33 LetGD GL .Q/, and letHD n2Z . ThenH is a subgroup of 2 0 1  50 G; in factH'Z. LetgD . Then 01         1 1 n 5 0 1 n 5 0 1 5n 1 g g D D : 0 1 0 1 0 1 0 1 0 1 1 1 HencegHg ¦H (andg Hg6H ). PROPOSITION 1.34 A subgroupN ofG is normal if and only if every left coset ofN inG is also a right coset, in which case,gNDNg for allg2G: PROOF. Clearly, 1 gNg DN ” gNDNg: Thus, ifN is normal, then every left coset is a right coset (in fact,gNDNg). Conversely, if the left cosetgN is also a right coset, then it must be the right cosetNg by (1.25a). Hence 1 gNDNg, and sogNg DN . 2 1.35 The proposition says that, in order forN to be normal, we must have that for all 0 0 g2G andn2N , there exists ann2N such thatgnDng (equivalently, for allg2G 0 0 andn2N , there exists ann such thatngDgn ). In other words, to say thatN is normal amounts to saying that an element ofG can be moved past an element ofN at the cost of replacing the element ofN by another element ofN . EXAMPLE 1.36 (a) Every subgroup of index two is normal. Indeed, letg2GXH . Then GDHtgH (disjoint union). HencegH is the complement ofH inG. Similarly,Hg is the complement ofH inG, and sogHDHg: (b) Consider the dihedral group n1 n1 D Dfe;r;:::;r ;s;:::;r sg: n n1 ThenC Dfe;r;:::;r g has index2, and hence is normal. Forn3 the subgroupfe;sg n 1 n2 is not normal becauser srDr s…fe;sg. (c) Every subgroup of a commutative group is normal (obviously), but the converse is false: the quaternion groupQ is not commutative, but every subgroup is normal (see Exercise 1-1). A groupG is said to be simple if it has no normal subgroups other thanG andfeg. Such a group can still have lots of nonnormal subgroups — in fact, the Sylow theorems (Chapter 5) imply that every finite group has nontrivial subgroups unless it is cyclic of prime order. PROPOSITION 1.37 If H and N are subgroups of G and N is normal, then HN is a subgroup ofG. IfH is also normal, thenHN is a normal subgroup ofG. PROOF. The setHN is nonempty, and 1.35 0 .h n /.h n /D h h n n 2HN; 1 1 2 2 1 2 2 1 and so it is closed under multiplication. Since 1.35 1 1 1 1 0 .hn/ Dn h D h n2HN20 1. BASIC DEFINITIONS AND RESULTS it is also closed under the formation of inverses, and soHN is a subgroup. If bothH andN are normal, then 1 1 1 gHNg DgHg gNg DHN for allg2G. 2 An intersection of normal subgroups of a group is again a normal subgroup (cf. 1.14). Therefore, we can define the normal subgroup generated by a subsetX of a groupG to be the intersection of the normal subgroups containingX. Its description in terms ofX is a little complicated. We say that a subsetX of a groupG is normal (or closed under 1 conjugation) ifgXg X for allg2G. LEMMA 1.38 IfX is normal, then the subgrouphXi generated by it is normal. 1 PROOF. The map “conjugation byg”,a7gag , is a homomorphismGG. Ifa2hXi, say,aDx x with eachx or its inverse inX, then 1 m i 1 1 1 gag D.gx g /.gx g /. 1 m 1 1 AsX is closed under conjugation, eachgx g or its inverse lies inX, and soghXig  i hXi. 2 S 1 LEMMA 1.39 For any subsetX ofG, the subset gXg is normal, and it is the g2G smallest normal set containingX. PROOF. Obvious. 2 On combining these lemmas, we obtain the following proposition. S 1 PROPOSITION 1.40 The normal subgroup generated by a subsetX ofG ish gXg i. g2G Kernels and quotients 0 The kernel of a homomorphism WGG is Ker. /Dfg2Gj .g/Deg: If is injective, then Ker. /Dfeg. Conversely, if Ker. /Dfeg, then is injective, because 0 1 0 1 0 0 .g/D .g/H) .g g/DeH) g gDeH) gDg . PROPOSITION 1.41 The kernel of a homomorphism is a normal subgroup. PROOF. It is obviously a subgroup, and ifa2 Ker. /, so that .a/De, andg2G, then 1 1 1 .gag /D .g/ .a/ .g/ D .g/ .g/ De: 1 Hencegag 2 Ker. /. 2  For example, the kernel of the homomorphism detWGL .F/F is the group ofnn n matrices with determinant1 — this group SL .F/ is called the special linear group of n degreen.

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