Transcription of Group Theory Notes
1 Group Theory Notes01234567 Donald L. KreherMarch 18, 2020iiAckowledgementsI thank the following people for their help in note taking and proof reading:Mark Gockenbach, Ryan McNamara, Kaylee Walsh, Tjeerd What is a Group ? .. Some properties are unique.. When are two groups the same? .. The automorphism Group of a graph .. more example..132 The isomorphism Subgroups .. Cosets .. Cyclic groups .. How many generators? .. Normal subgroups .. Laws .. Conjugation ..363 Even and odd .. Group actions .. Cayley s theorem .. The Sylow theorems .. Some applications of the Sylow theorems .. Simplicity of the alternating Group .
2 644 Finitely generated abelian The Basis Theorem .. many finite abelian groups are there? .. Generators and relations .. Smith normal form .. Applications .. fundamental theorem of finitely generatedabelian groups .. of Diophantine Equations ..835 A glossary of algebraic systems .. Ideals .. The prime field .. Algebraic extensions .. Splitting fields .. Galois fields .. Constructing a finite field ..976 Linear The linear fractional Group andPSL(2,q) .. conjugacy classes .. permutation character .. 1187 Automorphism Inner and outer automorphisms .. products .. ofSn.. 125 Chapter What is a Group ?
3 Definition :IfGis a nonempty set, abinary operation onGis a function :G G example + is a binary operation defined on the integersZ. Insteadof writing +(3,5) = 8 we instead write 3 + 5 = 8. Indeed the binaryoperation is usually thought of asmultiplicationand instead of (a,b)we use notation such asab,a+b,a banda b. If the setGis a finiteset ofnelements we can present the binary operation, say , by annbynarray called themultiplication table. Ifa,b G, then the (a,b) entry ofthis table isa in in: is an example of a multiplication table for a binary operation on thesetG={a,b,c,d}. a b c daa b c aba c d dca b d cdd a c b12 CHAPTER 1. INTRODUCTIONNote that (a b) c=b c=dbuta (b c) =a d= :Abinary operation on setGisassociativeif(a b) c=a (b c)for alla,b,c onZis not an associative binary operation, but addition + examples of associative binary operations are matrix multiplicationand function setGwith a associative binary operation is called asemigroup.
4 Themost important semigroups are :Agroup(G, ) is a setGwith a special elementeon which an associative binary operation is defined that a=afor alla G;2. for everya G, there is an elementb Gsuch thatb a= :Some examples of The integersZunder addition +.2. The setGL2(R) of 2 by 2 invertible matrices over the reals withmatrix multiplication as the binary operation. This is thegenerallinear groupof 2 by 2 matrices over the The set of matricesG={e=[1 00 1],a=[ 1 00 1],b=[100 1],c=[ 100 1]}under matrix multiplication. The multiplication table for this groupis: e a b cee a b caa e c bbb c e acc b a e4. The non-zero complex numbersCis a Group under WHAT IS A Group ?
5 35. The set of complex numbersG={1,i, 1, i}under multiplication table for this Group is: 1i 1 i11i 1 iii 1 i1 1 1 i1i i i1i 16. The setSym(X) of one to one and onto functions on then-element setX, with multiplication defined to be composition of functions. (Theelements ofSym(X) are calledpermutationsandSym(X) is calledthesymmetric grouponX. This Group will be discussed in moredetail later. If Sym(X), then we define the image ofxunder tobex . If , Sym(X), then the image ofxunder the composition isx = (x ) .) Exercises1. For each fixed integern >0, prove thatZn, the set of integers modulonis a Group under +, where one definesa+b=a+b. (The elementsofZnare the congruence classesa,a The congruence class ais{x Z:x a(modn)}={a+kn:k Z}.)
6 Be sure to show that this addition is well defined. Conclude that forevery integern >0 there is a Group Given integern >0 letGbe the subset of complex numbers of theforme2k ni,k Z. Show thatGis a Group under multiplication. Howmany elements doesGhave?4 CHAPTER 1. Some properties are (G, )is a Group anda G, thena a=aimpliesa= Gsatisfiesa a=aand letb Gbe such thatb a= (a a) =b aand thusa=e a= (b a) a=b (a a) =b a=eLemma a Group (G, )(i)ifb a=e, thena b=eand(ii)a e=afor alla GFurthermore, there is only one elemente Gsatisfying(ii)and for alla G, there is only oneb Gsatisfyingb a= a=e, then(a b) (a b) =a (b a) b=a e b=a by Lemma b= Gand letb Gbe such thatb a=e. Then by (i)a e=a (b a) = (a b) a=e a=aNow we show uniqueness.
7 Suppose thata e=aanda f=afor alla G. Then(e f) (e f) =e (f e) f=e f e=e fTherefore by Lemma f=e. Consequentlyf f= (f e) (f e) =f (e f) e=f e e=f e=fand therefore by Lemma Finally supposeb1 a=eandb2 a=e. Then by (i) and (ii)b1=b1 e=b1 (a b2) = (b1 a) b2=e b2= SOME PROPERTIES ARE :Let (G, ) be a Group . The unique elemente Gsatisfyinge a=afor alla Gis called theidentityfor the Group (G, ). Ifa G, the unique elementb Gsuch thatb a=eis calledtheinverseofaand we denote it byb=a >0 is an integer, we abbreviatea a a a ntimesbyan. Thusa n=(a 1)n=a 1 a 1 a 1 a 1 ntimesLet (G, ) be a Group whereG={g1,g2,..,gn}. Consider the multiplica-tion table of (G, ).gjgigi gjLet [x1x2x3 xn] be the row labeled bygiin the multiplication gj.
8 Ifxj1=xj2, thengi gj1=gi gj2. Now multiplying byg 1ion the left we see thatgj1=gj2. Consequentlyj1=j2. Thereforeevery row of the multiplication table contains every elementofGexactly oncea similar argument shows thatevery column of the multiplication table contains every ele-ment ofGexactly onceA table satisfying these two properties is called a Latin :Alatin squareof sidenis annbynarrayin which each cell contains a single element form ann-element setS={s1,s2,..,sn}, such that each element occurs in each row exactlyonce. It is instandard formwith respect to the sequences1,s2,..,snif the elements in the first row and first column are occur in the orderof this 1. INTRODUCTIONThe multiplication table of a Group (G, ), whereG={e,g1,g2.}
9 ,gn 1}is a latin square of side n in standard form with respect to the sequencee,g1,g2,..,gn converse is not true. That is not every latin square in standard formis the multiplication table of a Group . This is because the multiplicationrepresented by a latin square need not be :A latin square of side 6 in standard form with respectto the sequencee,g1,g2,g3,g4, above latin square is not the multiplication table of a Group , becausefor this square:(g1 g2) g3=g3 g3=ebutg1 (g2 g3) =g1 g5= Exercises1. Find all Latin squares of side 4 in standard form with respect to thesequence 1,2,3,4. For each square found determine whether or not itis the multiplication table of a If (G, ) is a finite Group , prove that, givenx G, that there is apositive integernsuch thatxn=e.
10 The smallest such integer iscalled theorderofxand we write|x|= LetGbe afiniteset and let be an associative binary operation onGsatisfying for alla,b,c G(i) ifa b=a c, thenb=c; and(ii) ifb a=c a, thenb= SOME PROPERTIES ARE (G, ) must be a Group . Also provide a counter example thatshows that this is false ifGis Show that the Latin Squareeg1g2g3g4g5g6g1eg3g5g6g2g4g2g3eg4g 1g6g5g3g2g1g6g5g4eg4g5g6g2eg3g1g5g6g4eg2 g1g3g6g4g5g1g3eg2is not the multiplication table of a :A Group (G, ) isabelianifa b=b afor allelementsa,b G.(a) Let (G, ) be a Group in which the square of every element is theidentity. Show thatGis abelian.(b) Prove that a Group (G, ) is abelian if and only iff:G Gdefined byf(x) =x 1is a 1.
