[PDF] GROUP THEORY (MATH 33300) 1. Basics 3 2. Homomorphisms 7 3

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GROUP THEORY (MATH 33300)

COURSE NOTES

CONTENTS

1. Basics 3

2. Homomorphisms 7

3. Subgroups 11

4. Generators 14

5. Cyclic groups 16

6. Cosets and Lagrange"s Theorem 19

7. Normal subgroups and quotient groups 23

8. Isomorphism Theorems 26

9. Direct products 29

10. Group actions 34

11. Sylow"s Theorems 38

12. Applications of Sylow"s Theorems 43

13. Finitely generated abelian groups 46

14. The symmetric group 49

15. The Jordan-H

¨older Theorem 58

16. Soluble groups 62

17. Solutions to exercises 67

Recommendedtexttocomplementthesenotes: J.F.Humphreys, ACourseinGroup

Theory (OUP, 1996).Date: January 11, 2010.

These notes are mainly based on K. Meyberg"s Algebra, Chapters 1 & 2 (in German). 1

2 COURSE NOTES

GROUP THEORY (MATH 33300) 3

1. BASICS

1.1.Definition.LetGbe a non-empty set and fix a map:GG!G. The pair

(G;)is called agroupif (1) for alla;b;c2G:(ab)c=a(bc)(associativity axiom). (2) there ise2Gsuch thatea=afor alla2G(identity axiom). (3) for everya2Gthere isa02Gsuch thata0a=e(inverse axiom) . is called thecomposition(sometimes alsomultiplication) andeis called the identity element(orneutral element) ofG, anda0theinverseofa. Where there is no ambiguity, we will use the notationGinstead of(G;), andabinstead ofab. We will denote byan(n2N) then-fold product ofa, e.g.,a3=aaa.

1.2.Example.The simplest examples of groups are:

(1)E=feg(thetrivial group). (2)(f0g;+g),(Z;+),(Q;+),(R;+),(C;+), where+is the standard addition. (3)(f1g;),(f-1;1g;),(Q;),(R;),(C;), wheredenotes the usual multiplica- tion andQ=Qn f0getc.

1.3.Lemma.Letabe an element of the groupGsuch thata2=a. Thena=e.

Proof.We have

a=ea(identity axiom) = (a0a)afor somea02G(inverse axiom) =a0a2(associativity axiom) =a0a(by assumption) =e(by definition ofa0):(1.1)

1.4.Exercise.Show that

(1) Ifa0is an inverse ofa, thenaa0=e. (2)ae=afor alla2G. (3) The neutral element ofGis unique. (4) For everyathere is a unique inversea0. We will denote it bya-1:=a0. (5)(a-1)-1=a. (6)(ab)-1=b-1a-1. We extend the definitionanto negative integersn < 0by settingan:= (a-1)-n.

We also seta0=e.

1.5.Definition.The number of elements of a groupGis called theorderofGand is

denotedjGj.Gis called afinitegroup ifjGj<1andinfiniteotherwise.

4 COURSE NOTES

1.6.Example.LetZndenote the setf0;1;:::;n-1g, wheremis the residue class

modulon(that is, the equivalence class of integers congruent tommodn). Then: (1)(Zn;+)is a finite group of ordern. (2)(Zn;), withZn=fm2Zn:gcd(m;n) =1g, is a finite group of order'(n)= the number of integers< nthat are coprime ton(Euler"s'function).

1.7.Definition.A groupGis calledabelian(orcommutative), ifab=bafor all

a;b2G.

1.8.Example.All of the above examples are abelian groups. An example of a non-

abelian group is the set of matrices (1.2)T= x y

0 1=x!

:x2R; y2R where the composition is matrix multiplication.

Proof.We have

(1.3) x 1y1 0 1=x 1! x 2y2 0 1=x 2! x 3y3 0 1=x 3! wherex3=x1x2andy3=x1y2+y1=x2. HenceTis closed under multiplication. Ma- trix multiplication is well known to be associative. The identity element corresponds tox=1,y=0. As to the inverse, (1.4) x y

0 1=x!

-1 1=x-y 0 x! 2T: ThereforeTis a group. It is non abelian since for example (1.5) 2 0 0 12 1 1 0 1! 2 2 0 12 6= 212
0 12 1 1 0 1! 2 0 0 12

1.9.Exercise.Prove the following:

(1) The set (1.6) SO(2) = x-y y x! :x;y2R; x2+y2=1 forms an abelian group with respect to matrix multiplication. (SO stands for "special orthogonal".) (2) LetKbe a field andKnnthe set ofnnmatrices with coefficients inK. Then (1.7) GL(n;K) :=fA2Knn:detA6=0g is a group with respect to matrix multiplication. (GL stands for "general lin- ear".)

GROUP THEORY (MATH 33300) 5

1.10. The easiest description of a finite groupG=fx1;x2;:::;xngof ordern(i.e.,

x i6=xjfori6=j) is often given by annnmatrix, thegroup table, whose coefficient in theith row andjth column is the productxixj: (1.8) 0 B BBB@x

1x1x1x2::: x1xn

x

2x1x2x2::: x2xn............

x nx1xnx2::: xnxn1 C CCCA:

The group table completely specifies the group.

1.11.Theorem.In a group table, every group element appears precisely once in ev-

ery row, and once in every column. Proof.Suppose in theith row we havexixj=xixkforj6=k. Multiplying from the left byx-1 iwe obtainxj=xk, which contradicts our assumption thatxjandxkare distinct group elements. The proof for columns is analogous.

1.12.Example.Consider a finite groupG=fe;a;bgof order 3. Ifeis the identity,

the first row and column are already specified: (1.9) 0 B @e a b a?? b? ?1 C A: If the central coefficient?is chosen to bee, then the ? below can, in view of Theorem

1.11 applied to the second column, only beb-but then there are twob0sin the final

row. Hence the only possibility is: (1.10) 0 B @e a b a b e b e a1 C A: We have thus shown that there exists only one group of order 3.

1.13.Example.LetG=fe;a;b;cgand assumea2=b2=e. Then the group table is

(1.11) 0 B

BBB@e a b c

a e?? b?e? c? ? ?1 C CCCA:

6 COURSE NOTES

The only possibility for?isc, otherwise there would be twoc"s in the last column. Hence (1.12)0 B

BBB@e a b c

a e c b b?e? c?a?1 C CCCA:

Again?must bec, and thus

(1.13)0 B

BBB@e a b c

a e c b b c e a c b a e1 C CCCA: Hence the group table is completely determined by the relationsa2=b2=e. The associativity of the composition law can easily be checked (this is a tedious but in- structive exercise). The resulting group is calledKlein four group.

1.14.Exercise.Write down the group tables for all residue class groupsZpfor all

primesp17.

1.15.Exercise.LetGbe the set of symmetries of the regularn-gon (i.e.,Gcomprises

reflections at diagonals and rotations about the center). Show thatGforms a group of order2n, if the composition is the usual composition law for maps. [This group is called thedihedral groupDn; we will meet it again later in the lecture.]

1.16.Exercise.LetKbeafinitefieldwithqelements. DeterminetheorderofGL(n;K).

GROUP THEORY (MATH 33300) 7

2. HOMOMORPHISMS

2.1.Definition.Let(G;),(H;)be groups. The map':G!His called ahomo-

morphismfrom(G;)to(H;), if for alla;b2G (2.1)'(ab) ='(a)'(b):

2.2.Example.

(1) Lete0be the identity element ofH. Then map':G!Hdefined by'(a) =e0 is a homomorphism. (2) The map exp:R!R,x7!ex, defines a homomorphism from(R;+)to (R;), since ex+y=exey. (3) The map:Z!Zn,m!m, defines a homomorphism from(Z;+)to (Zn;+).

2.3.Exercise.Show that

(1) the maps'1;'2:R!Tdefined by (2.2)'1(t) = 1 t 0 1!

2(t) =

et0 0e-t! are homomorphisms. (2) the map'3:R!SO(2)defined by (2.3)'3(t) = cos(t) -sin(t) sin(t)cos(t)! is a homomorphism.

2.4.Definition.A homomorphism':G!His called

(1)monomorphismif the map'is injective, (2)epimorphismif the map'is surjective, (3)isomorphismif the map'is bijective, (4)endomorphismifG=H, (5)automorphismifG=Hand the map'is bijective.

2.5.Definition.Two groupsG;Hare calledisomorphic, if there is an isomorphism

fromGtoH. We writeG'H.

2.6.Exercise.Show that(Z;+)'(2Z;+).

2.7.Exercise.Decide whether the homomorphisms in Exercise 2.3 are mono-, epi-,

or isomorphisms.

8 COURSE NOTES

2.8.Lemma.Let':G!Hbe a homomorphism, and lete;e0denote the identity

elements ofGandH, respectively. Then (1)'(e) =e0. (2)'(a-1) ='(a)-1. (3)'(an) ='(a)nfor alla2G,n2Z. [(1) and (2) are of course special cases of (3).] Proof.(1) We have'(e) ='(ee) ='(e)'(e)and (1) follows from Lemma 1.3. (2)'(e) ='(a-1a) ='(a-1)'(a), which proves (2) in view of (1). (3) follows from (1) trivially whenn=0, and by induction forn > 0. Forn < 0 '(an) ='((a-1)-n)(by definition) ='(a-1)-n(as we have just proved) = ('(a)-1)-n(by (2)) ='(a)n(by definition):(2.4)

2.9.Definition.Let'be a homomorphism from(G;)to(H;), and denote bye;e0

denote the respective identity elements. The set (2.5) im'=f'(a) :a2GgH is called theimageof', and (2.6) ker'=fa2G:'(a) =e0gG thekernelof'.

2.10.Exercise.Prove that(im';)and(ker';)are groups.

[We will return to this problem in the discussion of subgroups.]

2.11.Theorem.'is a monomorphism if and only if ker'=feg.

Proof.Assume'is injective. Ifa2ker', then'(a) =e0='(e)and hence by injectivitya=e. Conversely, assume ker'=feg. Leta;b2Gsuch that'(a) ='(b). We need to show thata=b. e

0='(b)'(a)-1

='(b)'(a-1)(Lemma 2.8) ='(ba-1):(2.7) Thusba-12ker', and hence, by our assumption ker'=fegwe concludeba-1=e, i.e.,a=b.

GROUP THEORY (MATH 33300) 9

2.12.Theorem.

(1) If':G!Hand :H!Kare homomorphisms, then so is ':G!K. (2) If':G!Hand :H!Kare isomorphisms, then so is ':G!K. (3) If':G!His an isomorphism, then so is'-1:H!G. (4) The identity map id:G!G,a7!ais an automorphism.

Proof.(1) We have

(2.8) ( ')(ab) = ('(ab)) = ('(a)'(b)) = ('(a)) ('(b)) = ( ')(a)( ')(b): (2) In view of (1) it remains to be shown that 'is bijective-this is evident and left as an exercise. (3) Letx='(a),y='(b), and soa='-1(x),b='-1(y). Now (2.9)'-1(xy) ='-1('(a)'(b)) ='-1('(ab)) =ab='-1(x)'-1(y): (4) This is evident.

2.13. The above theorem has an important consequence: IfG'Hthen by (3)H'

G. IfG'HandH'Kthen by (2)G'K. Finally, by (4) we haveG'G. Hence' defines an equivalence relation on groups. Recall: Given a setX, anequivalence relationRis defined as a subset ofXXwith the properties: (1)(x;x)2Rfor allx2R(reflexivity axiom). (2)(x;y)2Rimplies(y;x)2R(symmetry axiom). (3)(x;y);(y;z)2Rimplies(x;z)2R(transitivity axiom).

If(x;y)2Rwe sayxandyareequivalentand writexy.

2.14. Let AutGbe the set of automorphisms':G!G. Because of Theorem 2.12

(2) and (3) we find that if'; 2AutG, then' 2AutGand'-12AutG. (4) says that id2AutG. Hence(AutG;)is a group, called theautomorphism group ofG.

2.15.Lemma.Giveng2G, define the map'g:G!Gby'g(a) =gag-1. Then

g2AutG.

Proof.'gis a homomorphism since

(2.10)'g(ab) =gabg-1=gag-1gbg-1='g(a)'g(b):

It is in fact invertible:

(2.11)'-1g='g-1 since'g'g-1(a) =g(g-1ag)g-1=a, and so'gis bijective.

10 COURSE NOTES

2.16.Definition.'2AutGis called aninner automorphismif there is ag2G

such that'='g. Two elementsa;b2Gare calledconjugateif there is ag2Gsuch that'g(a) =b. We writeab.

2.17.Exercise.Show thatis an equivalence relation.

2.18.Exercise.Show that fora;b2Gthe elementsabandbaare conjugate.

2.19.Theorem.The map:G!AutG,a7!'a, is a homomorphism.

Proof.We have for any fixedg2G

(2.12)'ab(g) =abg(ab)-1=abgb-1a-1='a(bgb-1) ='a'b(g); so'ab='a'b, i.e.,(ab) =(a)(b).

2.20.Definition.The kernel ofis called thecenterofGand is denoted byZ(G).

Explicitly,

Z(G) =fa2G:'a=idg(by definition)

=fa2G:aba-1=bfor allb2Gg =fa2G:ab=bafor allb2Gg:(2.13) HenceZ(G)is the set of elements inGthat commute with all elements inG. Note that obviouslyZ(G)is a group, cf. also Exercise 2.10.

2.21. We haveZ(G) =Gif and only ifGis abelian.

2.22.Exercise.Determine all automorphisms of the Klein four group.

2.23.Exercise.Show that the symmetry group of a rectangle (that is not a square) is

the Klein four group.

2.24.Exercise.Set:=e2i=n, and letG=fk:k=1;:::;ngbe the group of thenth

roots of unity, where the composition is standard multiplication inC. (1) Show that':Z!G,m7!m, is a homomorphism. (2) Calculate ker(').

2.25.Exercise.LetGbe a group. Show that:

(1) If AutG=fidgthenGis abelian. (2) Ifx7!x2defines an homomorphism ofG, thenGis abelian. (3) Ifx7!x-1defines an automorphism ofG, thenGis abelian.

GROUP THEORY (MATH 33300) 11

3. SUBGROUPS

3.1.Definition.A non-empty subsetHGis called asubgroup, ifHis a group

with respect to the same composition as inG; we will write in this caseHG.

His called aproper subgroupifH6=G; we writeH < G.

3.2.Example.

(1)(Z;+)<(Q;+)<(R;+)<(C;+). (2) Ifd2Ndividesn2N, then(nZ;+)(dZ;+). (3) The groups in Example 1.8 and Exercise 2.3 are subgroups of GL(2;R).

3.3.Theorem.LetGbe a group andHGa non-empty subset. ThenHis a sub-

group if and only if (3.1)(a;b2H))(ab2Handa-12H). Proof.AssumeHis a subgroup. Then the image ofHHunder the composition :GG!Gsatisfies(H;H)H, i.e.,ab2Hfor alla;b2H. Ifeis the identity inH, we havee2=e, but this means by Lemma 1.3 thateis also the identity inG. Hence the inverse ofainHis also the inverse ofainG, and soa-12H. Conversely, assume (3.1). Then the compositiononG, restricted toH, yields a mapHH!H,(a;b)7!ab. The map is clearly associative (since this is true in the full setG), and we only need to show that the identityeinGis contained inH. But this follows from takingb=a-1in (3.1).

3.4.Corollary.LetGbe a group andHGa non-empty subset. ThenHis a sub-

group if and only if (3.2)(a;b2H))(ab-12H): Proof.AssumeHis a subgroup. Leta;b2H. Then, by Theorem 3.3,b-12Hand ab -12H. On the other hand, assume (3.2) holds. In particular (fora=e)b2H impliesb-12Hand hence(a;b2H))(a;b-12H))(ab2H)by (3.2). Thus by

Theorem 3.3His a subgroup.

3.5.Theorem.LetGbe a group andHGafinitenon-empty subset. ThenHis a

subgroup if and only if (3.3)(a;b2H))(ab2H): Proof.The first implication follows from the previous theorem. Hence assume (3.3) holds. SinceGis a group, for every fixeda2Gthe mapG!G,x7!ax, isquotesdbs_dbs12.pdfusesText_18

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