INTRODUCTION TO 'rUE THEORY OF FINITE GHOUPf3
UNIVERSITY MATHEMATIOAL TEXTS GlJlNERAL EDITOIlS
ALillXANDER C. AITKEN, D.So., .F.R 8. DANIEL E. RUTIIER}!'ORD, Dn. MATI!. DETERMINANTS AND MA'pnICES 8TA'XISTWAL MATllEMATlCS 'WAVES
.
ELJiWTnIOITY
•
Prof A Prof A Pl'of C A 2rof 0 A.
PIIOJEOTIVll GF-OMDTIW
C. Aitken, D Se , F.R S. C AItken, D.Se., F.R.S. ('oulson, D.Se., F.R.S E. Cuulson, D.Se., F H.B K . 'l' E. Fanllmer, Ph.D
R I'
lJo/TEClRA'l'j()N
Gille~pio,
Ph.D
Prof J ]\1 HYblop, D.Sr.
INFIlH'l'}: SmllIGS
INTEGlIA'rWN OF OImIJo/ARY D[F~'ImENTll\L EQUATION,>
E. L. Inee, D fle. IN'xnODuoTION ·xo 'I'HE THEORY
QE'
FINITE GnouPS
'V. Ledeunann, l'h D , D 8l' ANALYTICAL ClEO~LETRY OF 1'1l1Un~ Dn'll!:NsIoNS
Prof. W. U. M'Crea, Ph D. E. G PhJ!hpq, M.A , M Se D Eo Rutherford, Dr. M!tth. . Prof. H. W. Turnbull, FRS.
FUNOTIONS OF A COMPLEX VARIADLD VEO'roR ~IETIIODS THEORY OF EQUAnONS
Other volumea .n preparation
INTRODUCTION TO TI-IE TFIEORY OIt' FINITE GROTJPS
BY
W A LT.E::H r.a( j'UllIR
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Tjl~DBRMANN,
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AND BOYD
JoJIJINBURGII AND LONDON Nmw YOHK: IN'I'EHSmF!N('B rURLlHfIlmH, lNC.
HHII
PREFACE AUi'IIOU<.HI the Tlwory of UrOllp'l o(:('npies a positlOll of fundamentfl,l importarH't: HI !Hrllll'l'fl lilfti,h<.'mat,ir'R, it iR HUll cxcluderl from nIl HOnlllll'S emll'so ill, iL llum},r'I' of BritiRIl UnlverRit,ips. 'I'hi~ Htate of !d'fairH IR no doubt pRrtly r111<' io the fltd, that l'1l'llI('ntary CXpO'lttiollH of th€' Hubjr'('l nro not rCitlhIy aVaIlahle in Ellghsh, amI It iR II('IH>(I that thiy book will RllJlPIy tlw rcmedy I have, therefore, cmll'lwo\1l'('d tu III ilk!· tlli'l intl'{)(llll'twlI to thc Theory of GrOU!JH of Klleh all l'leltll'utary ('hnmntcl' tlw,t it can 1)0 renr1ily lInder~to()d hy an HOnOUl'R stllrlPHt ill second or third yenr. lVith tlm\ end in VleW I have never hesitated to oacrifice completeness for breadth 01' to reject more modern method" when 1 ('olll:lldereu altE'rnative presentations t.o 1Je more mtelll!.;ible :\Ittny WIll r('gret that the theory of matrix repn'Hentat.lOn has not heen mr:luded In my opmion, however, this extensive :mbJeut shoukll1ot be divoreed from its context Ul tile theory of linear associatIve algebras, and I felt that an adeqlULtc presentation of both of these disciplines was out of the qnestlOn m the space at my disposal My warm thankH are due to the Echtors for enroumging me to write thIS book, and C'spcemlly to ])1'. D R Hutherfol'd for the great care and the hf.'lpflll illtel'O~t WIth which he has followed its progress from the clay wholl the pInn WllS first discussed during a hohday in the Highlawls, Ilntil thc last proof sheet was retUl'nerl to the printers. I am indebtE'fl to my colleagun MI'. D Hc'cs for valtuthle suggestions and faT eheckin~ the elmmplr:-l, litH I to my wife for helping with the }ll'(Jof l'Pudmg and with tht, irlflex.
rns
v
PREFACE
vi
Flllally, I should like to express my appreciatlOn of the efficiency with which the pnLIishers have carried out their Lask under difficult conditIOns and of their never-faIlmg courtesy. W. LEDERMANN MANOHESTER
May HUB
CONTENTS OHAl'Trm r
TIfF.
mwup
CONCgp'L' l'MI~)
lllirodltchllll Z. Tho AXWlllq fir UrlJl1!' 'fht'l)l'J' a. K!
H ·k AHornlttlvn AXWlll'l fOl' Fimlo Gl'OUpH 5, ThQ Multlpli('[tj,llJn 'l'ltblo ll, EXlLmph'~ of FtnitO UrfHlpH
7, IH0I!10rpIIlC (kOtlpH 8. Tho Order (PerIOd) of un Blement 0. Cyolic Groups (,HAI'~'DR
1
"
7 H 11 It HI 21
21
II
cmlPLEXES AND SUBGROUPR 10 Tho Calculus of ComplexQ'l 11. Subgroup~ 12 Lagrange's Theorem 13. Subgroups of a Cvchc Group 14. The H.C.I<'. and the L.a 1\1 15. The DmlCt Product Ifi. Su.-voy of Gl1,)Up" up to Order 8 17 The Product 'fheol'mu 18 Decompoblttun rolahve to Two RubgrQups
CJIAPTJm 111
GROUPS Oli' PRR1\lU'l'ATJONS 10 Tho Symmotno Group Pn 20. Ull'cnlhw PermutatHHlH (l'ycl~\R)
vii
28 31 33 38 4.0 45 48 50 58
CONTENTS
VIII
PAGn
21. 22. 23. 24. 25 26. 27 28
Classes of PermutatlOns TransposltIOns Tho Altelllatmg Gral.IIl An Cayloy's Theorem TransltIVO Groups Pnrnitivo Groups Gnnom! Rnmmks about TransformatIOns nrollps related to Goometnca! ConfiguratlOns
70 73 76 78 82 85 87 88
('IIAI"rEJR IV
INVARIAN'r SUBGROUPS 2!1 ClnssoH of Conjllgatn r,j!oments Invariant Hllbg'I'OIlPS :11 Tho QlloUout (F!lCtOl) Group :l2. l.'ho Central 33 The Commutator G!OUP :l4 Homomorphlbills and I~oIllorplll",rnH 3;' Automo!plu'Ims 36 'Theorems on QuotHmt Groups 37 'rhe Jordan-Holder CompositlOn Theorem :l8. UalOls' Theormn on tho Alternatmg Group
aIJ.
013
00 102 103 104 1013 10\) 110 114 120
OHArTER V
SYLOW GROUPS AND PRIME POWER GROUPS 30. A Lemma on Abelmn Groups 40. Sylow's Theorems n Prime Power Groupb
126 127 134
CIIAI'TER VI
ABELIAN GROUPS 42. 43. 44. 45. ·16.
Inh /Iductron Sylow Groups of an Abelian Group BI1Si'l of Ahelmn (houps The Fundament,!! 'Theorem '1'110 !'~lGmental'y D,VIsor" and Invarlants of an Allf>1tarL Uroup
] 36 136 138 140
BllII.w(1lt ~l'l{Y
151 151
I!'
] 44
CHAPTER
I
THE GROUP CONCEPT 1. Introduction. The elementary 0PC'l'UtiollR of nrithnH'tie eOllfliflt in oornbiJllllg two IllUnh(Il'fl a alld h in ltc'('Ol'llallNl Wlf.ll Home wpll·!lefinPfl rllh'H NO It!! to pro(lllf'(' a 1I11iq1l0 tlurd numbm' c. l"or inRt,full'P, If the law of cOlUposition ordiwuy Illuitiplir:atiull. WP flhollid IllLvn (~-- (e/). Whf,1t a and b are given, til(' Il1I111b~r c can bo found III ctl,eh l'fJtlt'rcto
ue
CaRe. Rut we know that multiplIcation of two or lllore nllmherR obeys certam formal uw's wlueh hulll for all Jlrndllct~, IrrespectIve of their numeriC'al values, thuR : fib =ba (wmmlltalive hlW) (ah)c =a(hr) (as8ocwlH,e law) ](1
=a1 =a
(1.1) (12) (13)
The last equatiOn serves to introduce one particular nnmber called umty The second law states more explIcitly that, if wo put ab = Ii and be = t, then It is alw!tYs true that 8e = at. In the axiomatic t,reatmcnt of arithmetic It i'3 cnstomary to begin by htying down IJOstulatc'l or axioms such as (1 1), (1 2) and (1.3), and certain others dealing with addition afl well as multiplication, and then to deduce the logical con!1~· quenccs of thefle postulates It is immaterial, at the ontset, whether the Hymbnls a, b, . . . represent numherH a~ we normally understand them or other mn.t,hematical entiticH, or indeed whether they admit of any concrete interpretation at all. On the other hand, It will he conccdecl that it iii the variety and depLh of al'plieatioJl in pure and appl1erl mathe· 1
2
INTRODUCTION TO THE THEORY OF FINITE GROUPS
matics wInch has caused one conceivable system of axioms to be preferre(] to another 2. The AxiOInS of Group Theory. The abbtract theOly of groups deals with certain sets of elements G' A,B, 0, . .
with 1 CHpcci, to WhlOh a ,nngle law of comlJOszt1On IS defincd It is It matter of convention that the notatlOll and nomenclatllre of 'fmLltiplication are usuCLlIy adopted to express the eomposition of abstraet group elementfl. Tlll1S we aflfHlIUe that any two element:> A, B of G, equal or unequal, pOARess II umqlle product 0, and we write
AB=O
°
It iR the most typical l)l'operty of a group that this pro(luet is Itself one of its elcmonts or, as It is often exIWOi>Herl, that a group is closed wzth rC811f'ct to mult!pllcatwn. A complete sytltern of axioms which a set G mnst obey, If it 1::; to he a group, IS given III the following: DEFINITION 1. A set G of a finite or znfimte rt'urnber of elements, for which a law of composition (" multiplication ") is defined, forms a group if the followmg cond'itwns are sat!sfiell :
(J) Closure: to every ordered
* lJltir of elements A, B of
G there belol!f/s a mugue element 0 of G, written
°
"",AB,
u'hirh is called tl!l' 111 oduct of A (l,n(l B (II) Associative law: 1J A, B, 0 are any tMee elements tch~ch need not be distmct, then
oj G,
(AB)O=A(BO), 80
that rfther sid" may he denoted by ABC.
'" 1.1'. tho pmr>l A, Band
n, It are regarded /IS dl~tlllAt if A*R,
THE GROUP CONCEPT
(TIl) Unit elern.ent: G contain8 ar~ .f' elPinent I, crilled 01 Identity 81lch that fol' every element
the nmt elemcnt A ofG
AI=IA=A. (IV) Inverse or reciprocal elern.ent: corresponding to every element A of G, thc/'e e ri8t8 in G an * element A-1 stwh that AA-1 - A-1A =1
It will be (JIJ~ervcd tlHLt the'ic po~tUlatPH clo~e1y l'l'sNnhle thoHC which govern ol'llilU1l'y multlplil'ntwn, execpt t1mt Uw commutative law is not required to hold for groups. DBFJNJ'rlON
2. ,I
ljlOll]J
which luw lhp adlIitioIlCllpl'O}/ert!J
thllt fm every two of tts elellumts ~1B=BA ~II
called
m~
Abelian
t
(OJ cormnutative) gron]).
The waiving of the commutative law for groups in gcneml makeR it llcceHHary to (hstmgUlsh between the clements AB :LIllI BA, whICh are sometimei'! eallell the right. or left· hand prodllctR of .A by B. It is (}lute poss1ble th:Lt whIle the commutathTe law docs not hold throughout the group, It may yet be valul for eel' Lam indivicluall)airs of elemenbl DEFINITION 3. 'l'wo clements A, B we 8aid to rom mute (or to be co:m:mutative, or per:mutable) if
AB=BA. It is worth winle to dwell a little longer on the SIgnificance and lUlUlCdi:Lte consequencefl of the group axioms stated in Definition 1, p. 2. '1'he fl880Ctativ(>, law wat; enlluoiatm[ only for three element:;, hnt can in fact he cai'lily proved in Its more • !La
umqllonos~
will be pravDu laLor on, seo pp.
+ After
N. II. Abel (1802-20).
fj
and 6.
4
INTRODUCTION TO THE THEORY OF FINITE GROUPS
general form, namely that, wIthout ambiguity, (A I A 2
•••
A r )(A r+1' .•. A,,) = (A I A 2 ••• A,)(AS+l ... An),
(14)
where rand 8 are any two integers between] and n Since when n = 3, (I 4) is eqUIvalent to axiom (II), we have a basis of mathematical mductlOn with respect to n, the number of factors. There is no loss of generality m assUlllmg that 8
••
A r = (A I A 2
.,
As) (As+! ... A r ),
and the left-hand sIde of (1.4) becomes
L(A I
•..
As)(A s+! ... A,)](A'+l ... An) = [B I B 2]B a,
where the products m round brackets have been denoted by B I , B 2 , B a respectively. Agam, the right-hand side may be expressed as (AI' .. A s)[(As+ 1
• ••
ArHA r +! ... An)] = B I (B 2B a].
In virtue of axiom (II) [B I B 2]B a= B I [B 2B u],
whlCh proves the prOpoAltion (14). We are t,herefore entitled to omit the brackets altogether and denote oj ther side of (1.4) by A I A 2 • •• An· In particular, when all factors are identIcal we shall, as in ordinary algebra, write AA =A2, (AA)A =A(AA) =Aa,
Also, when nand m are positIve integers, AmAn=AnAm=Am+n
(I 5)
and
(1 6)
THE GROUP CONCEPT
5
It IS interesting to note that the familiar law of indIces and the manipulatlOn of powers of a single quantIty finds its ultimate justification m the as'30Clatlve law of multIplIcation. When A and Bare distmct elements, we have in general
(AB)n*AnBn; but when A and B commute, (AB)n=ABAB . . AB=AnBn
(1.7)
and A711Bn=BnAm, since we may rearrange the order of the factors as we please. MultiplIcatlOn by the umt element I, whether used as a right or a left factor, leaves every element of G unaltered It follows that there can be only one such element :For supposing that J had the same propel'ties as 1, let us consluer the product
IJ. Since premultiphcation by 1 leave8 J unaltered, we have IJ =J; on the other hand, as pORtmultiphcation by J has no effect on I, it follows that IJ =1, hence
J =1J =1, which proves the uniqueness of the unit element. Also,
(1 8) The eXIstence, within the group, of an ~nve1'8e for every element A means that not only multiphcatioll, but also left and right" divIsion", can be carried out Wlthin a group. Thus if A and B be any elements whatsoever, there eXIst elements X and Y such that AX=B,
In fact we can express the solutlOns
X =A-IB,
(1.9)
YA=B III
the form
Y =BA-l,
6
INTRODUCTION TO THE THEORY OF FINITE GROUPS
using tho element A -1 referred to III postulate (IV) on p 3. Tlus argument also proves that the solutions of (1.9) are unique, for if it would follow that
A-1(AX I ) =A-1(AX 2 ), and therefore Xl =X2 • Again, since A -1 itself is a solutlOn of these equatlOns when B =1, we infer that the inverse of any given element is ~tni([1te.
It IS impOltant to note that A commutes with A-I, Hence by (1.8) and (1.7) I =1n = (AA-l)n =An(A-1)n,
so that (A-l)n is the inverse of An; it is customary to write . (1.10)
The reader will have no dIfficulty in convincing himself that the rules (1.5) and (1.6) are still valid when m or n are negative integers or zero, provldetl we put . (1.11)
In particular we observe that two powers of the same element always commute, thus
AkA! =A!A't,
(1.12)
Since, for any two elements A and B, (AB){B-IA-1) =ABB-IA-l=],
we have and more generally, (AB ... l[)-l=](-l,
,B-IA-l,
(1.13)
THE GROUP CONCEPT
7
Lastly, we remark that I IS the only idempotent element of the group, I.e. the only solution of the equatIOn X2=X.
(1.l4)
For on multIplying (1.14) by X-I we obtain X- IX 2=X-IX, 1
e
X =I.
3. Examples of Infinite Groups. The groups we arc going to mention in this sectIOn contain an infinity of elements and ~tre therefore called InfinIte groups. (1) All posittve latwnal mm~l;ers form a group with lespect to mchnury nmlttphcatwn Indeed the product of two
ratIOnal numbers IS a ratIOnal number, the urut element is the rational number 1, and the inverse of every positive ratIOnal number is also such a number. This group IS AbcUan. By way of contrast we might pomt out that the set of p08ttive tntegers does not form a group WIth respect to multiplIcatIOn, because the fourth postulate, the eXIstence of an inverse, is not fulfilled. (Ii) The set of all intege1s fm ms an Abelian group with respect to addition. In thIS case It IS customary to express the compositIOn of elements by such an equa,tion as a+b =0.
The unit element IS the number 0, because a +0 =0 +a=a,
and the lllverse of a is - a. (111) Rotatwns about a fixed point: If a rigid body is free to move about a fixed point 0, every displacement of the body is equi.valent to a rotatIOn through an angle IX about a lIne l passing through O. Such a displacement will be denoted by (l, IX), or more brIefly by a single letter A. If B is another dIsplacement of the same kind, the product
8
INTRODUCTION TO THE THEORY OF FINITE GROUPS
AB is defined as the displacemont which rcsults when A and B are carried out In succeSSIOn (in tIns ordcr'l<). ThIS being the law of COml)osition It IS 'easy to verify the group postulates The umt element is the" dIsplacement" which, In fltct, leaves the body In its anginal posItion, i.e. 1= (l, 0), where l is arbItrary The mverse of (l, IX) IS (l, -IX). We shall later (p. 87) prove thc associatIve law ror a certam general class of operatIOns which includcs thc present type of displacement The following Illustration demonstrates that the commutative law IS not always fulfilled' let 1234
~
y
m'~', 'm .'/1
y
I
4
3
4
2
3
2
1
1;
3
4
I
m
A
l
,
B
-qr
AB
BA
FIG. 1
denote a square lamina Imtially placed in the (x, y)-planc as indICated In fig. 1, the axIS of z bem/; at right angles to the plane of the lamina. We assume that Oxyz IS a righthanded system of reference, which is fixed In space. If, in the above notatIOn, A=
(z, i),
B = (x, 7T),
~.
it is easily verified that AB and BA give rise to different pOSItions of the lamma so that AB =,6 BA
* Some authors take the reverse order
THE GROUP CONCEPT
9
(iv) GrO~tps of mat1'ices. the reader who IS familiar with the elementary properties of matnees * wIll appreCIate the excellent lilustration of group theory afforded by matrix algebra and in partICular by matnx multiplicatIOn For the dlseusslOll of the assoCIative law, the existence of the unit matrix amI the reCIprocal matrix, see op cit p. 12 and p. 53 respectively, or Ohapter III, § 27 of the present book The following sets of matnces form groups with respect to matnx multIplication.
(a) all rzon-s~ngular matrices oj a fiJ.:nd order 11, (b) all orthogonal matrices of a fixed 01 der n, (c) all orthogonal matnces of a .fixed ordel' and of detel'mmant +1.
4. AlternatIve AxioIIlS for Finite Groups. Groups wInch consIst of a fimte number of elements aro of espeCIal mterest, and we shall henceforth be concerned only with :fmite groups unless tho contrary IS expressly statefl. 'l'he number of elements m a finite group is calleel the order of the group. In the case of finite groups the fundamental aXIOms (p. 2) may be replaced by the following alternative system of postulates WhICh are sometimes eaSIer to venfy: THEOREM I A fintte set of elements, for which a law of compositwn (" multipl2cation ") is defined, forms a group ~f
(a) the set is closed with respect to multiphcatwn ; (b) the assocwtive law ~s satisfied, (c) right and left cancelhng is permitted, i.e each of the
equations
AX=BX and
YA=YB
~mplies that
A= B Proof We shall show that condItion (c) entails aXIOms (III) and (IV) on p. 3, I.e. we shall cleduce from (c) the ,j<
See A. 0 AItken, Deterrmnant8 und
Matl'~Ce8, 4th
OdlLlOn. B
10
INTRODUCTION TO THE THEORY OF FINITE GROUPS
existence of the unit and mvcrse clements.
Let . (1.15)
be the g dIstinct elements of the group G aile at them, the products A.A 1 , A.A z, .
, AA{/
If A be any (l 16)
are elements of G m virtue of (a) and are dlstmet, because, by (c), an equation of the form AAt=AAJ imphes that At =A J Thus the g elements written down m (1 16) are merely a different arrangement of those m (1.15). Therefore if X be any element of G whatsoever, there exists an element Au such that (1 17) AAu=X, because X, WhICh oecms in (1 15), must have a counter. part in (1.16) Similarly, 011 posLmultiplymg tho set (115) by A we conclude that there e~is1,s an element Au such that (US)
These concJuflions apply III particular to the case when A =X, i e we infer that G contains two elements Ao and A o' such that (1 1!J)
and Ao'A =A
(1.20)
On premultiplymg (1 17) by A o' we obtain A o' X = Ao'(AA u) = (Ao'A)Au=AAu=X, ie Similarly,
Ao'X =X.
(1.21)
XAo=X.
(122)
Thus we have shown that in equations (1.19) and (1 20) A may be replaced by X which, it will be remembered, was an arbitrary element of G, i e any element whatsoever remains unaltered when premultiplied by A o' or post.
THE GRO\JP CONCEPT
II
multiplied by A 0' In particular, on putting X equal to A o m (1.21) and. X equal to A 0' in (1 22), we find that
Ao=Ao'Ao=A o' This establIshes the existence of a unit element, namely I =A o =A o' ;
Its uniqueness follows at once by an applicatIOn of (c) to the equation AI =A. Let Ufl now return to equations (1.17) and (1.18) and consIller the cai:!e m which X =1. We conclude that G containi:! two clementi:! A, and A/ i:!uch that
AAr=I
and
A/A =1. In order to prove tha,t these two clements are equal we sImplIfy the product A,'AA r in two different ways by appealing to the associative law, namely A/AA, and
=
(A/A)A, =IA,.=A r
whence
Ar=Ar'=A-I, say. The uniqueness of A -1. follows from the fact that, according to (c), there cannot be more than one solution of the equatIOn AX=I. 5. The MultlplicatlOn Table. In the abstract theory of groups whon no reference IS made to the nature of the elements, the group must he considered completely I:lpeClfied If all pOSSIble produotl:l AB are known or can be determined For a fimte group of order (/ there are (12 such products, which may conveniently be listed mag x (J :multiplication table as was suggested by A. Cayley.*
* Ph~l
Mag vol,
VH
(4), 1851-.
12
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Example. We shall gIve a detaIled di~cUfJfllOn of a group of order 6 which may in many ways be rega,rded as tYPICal The elements WIll be denoted by I, A, B,
a, D, E,
. (1.23)
and the 36 products formed by them are set out in the following table: I A E
a
D E
I
A
B
I
A B
B
A B
I .d
a a ]iJ
D
E
D
E
a
D E a D E I A D E C B I E a D A B J
D
a
E
A J
Table 1
For example, the product aD IS that element whICh stands at the intersection of the roW marked a and the column marked D, thus CD =A. SimIlarly, we find that BA =1, EE =D and so on. Our task is to verify that the rules for composItIOn summarIzed in Table 1 do m faot obey the group a,xioms laid down in Theorem 1 011 p. 14. the system (1.23) is obviously closed (postulate (a)) because the product of any two elements is agam one of the six elements of the set. The cancelling rule (c) is also fulfilled, for if U and V be dIstmct dements of the set, then XU 4'XV whatever element X may be chosen; this IS seen at once from the table where the varlQUS elements X U occupy the row marked X, and inspeotIOn shows that the elements in each row are dlstinot. The dIrect venfioatIOn of the assoc~attve law (b) would be rather laborIOUS as all pOSSIble equations of the forID (XY)Z=X(YZ) . (1.24) woulel have to be examined ; however, we bhalliater learn how this difficulty can often be overcome by indirect
13
THE GROUP CONCEPT
methods, and we shall then return to this example (p. 70). In the meantIme the reader is recommemled to check equatIOn (1.24) at least in some particular cases, e.g (AO)D=ED=B, A (OD) =AA =B.
Table 1 is not symmetrical wIth respect to the main diagonal, which Illustrates the fact that in general XY =F YX. But it will be observerl that the element [ always lies either on the diagonal or else occupies positions whICh are symmetl'lcally placed in pairs ahout the diagonal. This l11uHtrILtel:! the property that every clement commutes WIth its inverse, e.g AB =BA =1, showing that A and B are lllverse elements of each other The rOWH and columns, mc1uding then' hea,dings, may be rearranged amongst themselves without affectmg the mformatlOn contained III the multIplICation table. It IS sometimes converuent to specIfy the rows not by the elements themselveH but by their inverses, thus Table 1 is equivalent to --
1 A-I B-1 0-1 D-l ]§-~
'J A B C D jjJ ---------D E 1 A B B I A- D E A B I E 0 D 0 D E 1 A B D E 0 B 1 A ]§ 0 D A B 1
a
a
Table 2
In a multIplICatIOn table of this type the Ulllt element naturally occupies all the positions of the main diagonal Postulate (c) on p. 9 IS eqUIvalent to the demand that no element be repeated in anyone row or column of the g x g muitiphcatIOn table, I.e. aU (J elementfl appeltr in some order m each rOw 01' eolumn Such all arrangement is commonly ca.Jled a latin 8q~wr(j. But it is not true that
14
INTRODUCTION TO THE THEORY OF FINITE GROUPS
conversely every lat'In square can be lllterprotocl as the multiplication table of a fiuite grolll) , for while the
ABO
D
IIABGD A AID B 0 BBGIDA
o
0
D
D
D
BOA
AlB 1
Table 3
the break-down of the associative law is exemplified by the equations (AB)O =DO =A, A(BO)=AD=O, ThuA Table 3 cannot be regarded as the multIplication table of a group 6. Exam.ples of Finite Groups. (i) The numbe1's 1, i, - 1, - i form an Abelian (lrotl)! oj order 4 tmth 1 espect to ordina1 y m'ultiplicatton (i = V -=1). The reader is recommended to construct the 4 x 4 multiplication table (ii) Consider the six fUllctions J1(Z)=Z, f4(z)
=z'1
which are the values of the cross-ratio of fonr pomts when these points are permuted In all possible waYA, and let the law of composition be defined as 8ubshtut'Wn of one function m anothe1'; c g,
THE GROUP CONCEPT
15
z
I
141a =Mla(z)} =fa(z) = z -I =10' and so on. It wIll be found that these functions form a closed set, the part of the ~tmt clement being played by 11 (z). The multIplIcatIOn table IS readily constructed thus:
11 f 2 1a J4 fG - - - - - - ----11 fl 12 fa 14 1:; 12 12 fa 11 fo 14 la .fa fl 12 1s 10 J4 14 fG fo fl f2 fs 15 fo 14 1a fl
16 10
----~
10 la
],1
15
12
la
fG f4 fa f2
11
TablE' 4
The associatwe law can be verified or established by indirect arguments (Chap III, § 27). (in) Classes of residues. Every positIVe integer (> 1) gIves rise to a certl:tin finite Abelmn group m the following way' let m be a fixed positlve integer (> 1), which will henceforth be referred to as the :modulus. Two tntegers x and yare satd to be congruent to each other with regard to the mod~tl~ls m tl m Ul a factor of x - y, we write symbolically x==y (mod. m). Thus two integers are congruent with respect to m if and only if they differ by an exact (positive or negative) multiple of m; e.g 3 == 18 (mod. 5), - 2 == 14 (mod 8), 12=0 (mod. 3) It follows that any mteger whatsoever is congruent, with regard to the modulus m, to one and only one of the numbers . (1.25) 0,1,2,. ,m-2, m-l, which are therefore saId to form a co:mplete set of residues relative to m, they are m fact the least non-negative residues relative to m.
16
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Bearmg m mind that the congrnenue relatIOn x-=y (mod m)
is eqUIvalent to an equatIOn of the form x=y+km
where le is a certam integer, it is efLflY to verify the followmg rules: if Xi -= Yl (mod. m) and X2 = Y2 (mod. m), then c1X 1 +C2X2~C1Y1 +C 2Y2 (mod. m),
(A) where
C1
and
C2
are any integers, and X1X2~Y1Y2
(M)
(mod. m)
l'hese rules show a remarkable resemblance to those for ordmaryequations Yet the following important difference should be carefully noted. wIllIe the equation lex = ley implies tha.t, x=y provided k*G, we ca.1\. only BtU,te that lex:;:=ky (mod. m)
(D) ~rnplil3s
. (126)
that x~y
(moel m),promded* (m,le)=;1
Indeed (1.26) is equivalent to the statement,",
I
m k(x-y),
and if m is pnme to le, it follows that mlx-y,
ie
x~y
(mod m)
'l'ke number of integc1'8 between 0 and ill which are prime to ill is denoted by ep(m) (Euler's function). Thus, ep(O) denotes the number of mtegers between 0 and 8 which are
* "Ve use the symbol (a, b) to denote the highest common factor of a and b; III pattwular (m, k) =;= 1 means that m and lv are (relatlvcly) prime. t The symbol a I b expresseE, bhat a is a factor of b, I e. that there eXists an integer q such that b=aq. We shall use the fact that If a I bo and (a, 0) =1, then a lb.
THE GROUP CONCEPT
17
prlme to !J, there are evidently SIX such integers, namely 1,2,4,5,7,8 and so q;(9) =6 If p be a prime number, all but the first of the integers 0, 1, 2, ... , p - 1 are prime to p, whence q;(p) =p-l
. (1 27)
Again, if m =pr, only the multIples ofp in the set 0, 1, 2, ... , pr -1 are not prime to pr; as there are evidently p' -1 such multiples, namely hp where h = 0, 1, 2, .. , pr- 1 _1, it follows that q;(lf ) =p' - p'-l (1.28) It is customary to put ¢(1) =].
(1 29)
Those of the numbers (1.25) which are prlme to rn form a sub-set M which we shall denote by . (1.30)
where 0 < ai < m. One of these numbers, say all is equal to 1. The product of any two is a number which is certainly prime to m though it may be greater than m and therefore not mcluded m (130); but it is m any case congruent to one of these resIdues, as is mdeed any mteger whICh is prIme to m. Thus we may write
a,aJ..=al (mod. m)
• (1.31)
and define a law of composition for resIdues as mult~plication followed (If necessary) by positwe restdue w~th regard to m; e.g.
4 x 5=2 (mod 9),
O1d~nary
redttction to the least
2 x7=5 (mod. 9) etc.
It is clear that thIS law of compositIOn, being derived from ordmary multiplication, IS both commutative and aS8ociatwe. Agam, as we are concerned only with residues prime to m, Rule (D) (p 16) may be applied, I.e. If a, b and x be any members of M, then
ax""bx (mod. m) implies that a=b (mod. m).
IB
INTRODUCTION TO THE THEORY OF FINITE GROUPS
ThIs establishes the canccll~n(J t'1tlc (c) of 'rheorcm 1, p \:l Thus we can summarIze our results III the THEOREM 2 The least positwc ?es!d1tes pnme to ill form an Abelian groulJ M of order rp(m) ~f the law of cornJJos~tion be defined as m1lltiphcation followed by t'eductwn relative to the mod1tlus m The unit element of M is obvIOusly equal to 1, which always occurs III the Ret (1.30). Also since M IS a group, every element a of M pOSHc"ses a ulllque lllverse a' such that aa' "" 1 (mod m),
or expressed as an ordlllary equatIOn aa'=l +xm,
where x IS a certam llltcger to m, it is of the form b=a
. (132)
If b be any number primo j
yrn,
(133)
where a IS a suitable element of M. On llmltiplymg (1.33) by a' and substItuting for an' trom (1 32) we get a'b-(a'y+x)m=l
Tins ill a well·known result,"" namely that if band mare relatwely pnme, then integers u and v exist su,ch that ub +vm=l , III
. (1.34)
our notation 1!=a' and v=a'y+x.
7. IsornorphlC Groups. On closer scrutmy It will be observed that Table 1 (p. 12) dIffers from Table 4 (p. 15) merely m respect of notation, so that from an abstract point of view they are regarded as representing the same group. In fact, if we establish the (1, I)-correspondence
* The aI'gument :m the text is not offered as a proof of (L34), wInoh IS usually obtulUed by means of Euchd'J', algonthm, and In its turn forms the basis of rule D.
THE GROUP CONCEPT
k<-+I,
h·~--+A,
19
f3+-?-13, fo++E
f4 +-?-O, k«·-~D,
. (1 35)
the identIty of the tables becomes evident, i.e any relatIOn denved from Table 4 is in virtue of the correspondence (1 35) earned over into a correct relatIOn between the elements of Table 1, and VICe versa, e g,
f J3 ==.jo
OB =E
becomes
This is an IllustratIOn of a concept whieh is of fundamental importance in the theory of groups DmFINITION 4, '1'wo (frottJlB G = (A, B" ,} a'ltd G' ={A', B',. } w'e said to be isomorphic * if a (1, 1)c01'respondence A+->-A', B-<.->-B'",
can be
establ~shed
between their elements Buch that
AB=G and mce ve1'sa,
01'
~mph(>,s
more briefly
A'B'=O',
'1,1
(A13)' =A'13'
Paraphrasing this defimtlOIl we may say that tsomorphic lwve the same structure although thell may (hffe1' tn respect of the notation and natll/e of thell' elements Example 1. The following grou-ps of order' '" are ISOmorphIC, the law of oomposition for eaoh being stated 111 brackets:
(JlotiJiS
(G(.) the numbers ((3) the mat? ices
(y) the residlles
1, 1:, -1, - i,
(ordtna1 y multiplication) (matrix mttltiplication)
1 ] [-1 0 ] [ 0 -1 ] 0' 0 -1 ' _ ] 0 1, 2, 3. 4 (mod. 5) (mtlltiplication and 1'eduction mod. 5)
* Some uut.hors Ube t.he expreSSlOn " simlJly connectlOn, see foot.not.e on p lOG.
~801nol'Phia l>
m this
20
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Indeed if the elements of each set be renamed J, A, B, 0 (m thIS order), their common multIplIcation table is seen to be
I I
A B
o
ABO
ABO ABO I B 0 I A 0 I A B I
Table 5
E.g the abstract equation
OB=A is interpreted m (ct.)
as
(-i)(-l)=i,
(fi)
as
[
111
in (y)
~ -~
I -~ -~ J
=[ -
~ ~
J
as 4 x 3 ",,2 (mod. 5)
In a similar manner all other properties of the abstract Table 5 are reflected III the concrete groups (ct.), (fi) and (y). Example 2. OonsIder tho followmg groups of order 4 where in each case the law of composition is stated in bro,ckets: 1 I (substitution, see p. 14) (a) the Junctions iZ, -z,
z'
-z
(matnx multiplication)
(b) the maM'ices
[
~ ~
]. [
~ -~
J,
[-~ ol ']
(0) the residues 1, 3, 5, 7 (mod. 8)
[-10-10 ]
(multiplicatwn and reduction mod. 8)
If the elements of each set be denoted by I, A, B, 0, it will be seen that they have the same multiplIcation table, nllmely
THE GROUP CONCEPT
I A B C
I
A
B
0
I A B 0
A I
B
C
B C I
B
A
21
0
A I
Table 6
It is left to the reader to verify thIS statement. We would pomt out, however, that the groups represented by Tables 5 and 6 respectively arc certamly not Isomorphic j in the latter the square of each element is equal to I, which is eVidently not the case 111 Table 5. Incidentally, we have learned that groups of the same order may well be of different structure. 8. The Order (Period) of an Element. Let A be an element of a group G of order (J and consider the set of powers of A, I,A,A2,A3, ... , . (136)
all of which are of course elements of G. Since G is a finite group, these elements cannot all be distinct, 1 e. we must have a relatIOn Ak=A!, where k>l, say.
Hence
A7,-I=I, whlCh shows that in a finite group some power of every element is equal to the unit element. DEFINITION 5. 'Phe least positive power of A wh~ch is equal to the umt element is called the order (or period)
oj A. Thus rl A is of order h, then
Ah=], but when O<x
12.
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Again, If m be a multIple of h, Hay m =hq,
we have
Am= (Ah)blq=I
The converse of this proposItion is also true, shall prove
111
fact, we
THEOREM 3. If A is of order h, then Am = I if, and only if, m ~s a multiple oj h. Proof. DIvide m by h and let q be the quotient and r the remainder, thus m=hq+r, where . (1 37) O<'(r
I =Am =AhHr = AhqAr = IqAr =A'.
Rince h IS the order of A, this equation is ImpossIble unless 0; i,e. rn mUl:Jt be a multiple of h. The following facts about the order of an element are frequently used. (i) The umt element I ~s the only element of mde?' one, (il) The elements A nnd A-I are always of the same order (see (IV) (iii) If B =P-IAP where P is any element whatsoever, then A and B are of the same ordM'. For B2 = (P-IAP) (P-IAP) =P-IAIAP =P-IA2p,
l' =
and generally,
B"=p-lA'"P, AI'=PB'"P-I; thus if A"=I, we have B"=P-IIP=I, and vice versa. (IV) The mde?' of any power of A cannot exceed the order of A. For if Ah=] andB=As, thenBh =Ash= (Ah)2=P=I. Moreover, if A be of orde?' h and if s be prime to h, then As and A are of the same order. As (s, h) = 1, we can find two integers u and v (see p. 18) such that us +vh = 1 Hertce if B=As, we have
THE GROUP CONCEPT
23
BU =A'tB =A I-v" =AA -'I)" =AI =A. Therefore, A is a power of whence A and B are of the The following theorem clement of composite order
B wInch IS itself a power of A, same order. allows us to resolve a given into factors of simpler orders:
THEOREM 4. If C be of order mn where m and n are relatwely prime, then 0 can be expressed ~n one, and only one, way as the product of two commutat'ive elements A and B of orders m and n respectwely. Proof I Let A = On and B = om; the elements A a,nd B commute because they are powers of the same element O. We have whence we Imer that m and n are the orders of A and B respectively; foJ' if a smaller (positive) power of A 01' B were equal to the umt element, we should get a contradlOtion to the fact that 0 was of order mn Since m and n are relatIvely prime, we C!1D find integers u and v (see (1.34)) such that un+vm=1. . (1.38) Hence C =O"n+ ll m= (on),,(cm)V =A"Bv. We deduce from (138) that u is prime to m; for if any factor (> I) were common to u and m, it would divide eitch term on the left-hand side of (1.38) and therefore also the right-hand SIde, which IS a.bsurd. Hence by proposition (iv), p. 22, A" is of order m. Similarly, Bv is of order n Also, since A and B commute, so do A'l a.nd Bv Thus If we put J11 =A", N =BV we have
O=MN, • (1.39) where J11 and N fulfil all the conditions of the theorem. 2. In order to prove the uniqueness of the decomposition (1.39) suppose tha.t ... O=MN=M1N 1, • (1.40)
24
INTRODUCTION TO THE THEORY OF FINITE GROUPS
where M commutes with N, and lVI l wit.h N 1 , and where tho orders of 111 and 1I11 are ?n, and those of Nand N 1 are n Raismg (1.40) to the (n'lt)tll llower, we get lIInuNm' =1I1 nuN nu, 1
1
lYinu = M 1 nu, whence by using (1.38)
1I1 1 -
=lvI1 1-mv, M(llI?l1)-V =.M1(1l1 1"')-V, MI-v=M 1I-v, vLV
i.e
M =JJ11 Equation (1.40) now at once shows that, likewise,
N=N 1 9. Cyclic Groups. DE]'INJTlON 6. A grmtp whose elem.ents can all be e1:pressed as powers of a smgle elr-?nent is called a cyclic group.
The general form of a cyclic group of order c 18 C: I, A, A2,.
, A 0-1,
• (1.41)
where c is the least positive integer such that
AC=I. We say that C IS generated by A The order of a cyclic group IS equal to that of Its generating element, conversely, ~f a (li'OUp of order c contain an element of order 0, then the group ~s eycl~c. The generatmg element is not umquely determmed, mdeed if e be any integer prIme to c and O<e
25
THE GROUP CONCEPT
may be seen by mailing theIr generating elements correspond to one another; there ~8, in fact, one, and only one (abst?'act), cyclic group for any gwen 01 de?' All cyclic groups are Abehan. Example 1. If y=e 2rrt!0 (a primitive cth root of umty) , then the numbers , "1 0-1 form a cychc group of order c WIth respect to multiplication, because c is the least posItive exponent such that yO = 1. Example 2. The residues
..
0, 1,2,
.. , m-1
with respect to the modulus m form a cyclic group of order m, if the law of composition be additIOn followed by reductIOn to the least non.negatlve resIdue relatIve to m ; eg,ifm>J (m-2) +4=m +2=2, (m - 3) + 3 =m 0. The operations I, A, A 2, ••• , A 0-1 of a cychc group may be interpreted geometrically as rotations in a plane about a fixed point 0 The group is generated by the rotatIOn through an angle 271/C, any point on which this operation IS carrIed out c times in succession, just describes a complete circle (hence the name" cyclic" group).
=
ExaInples (I) Prove that the followmg sets of numbers form infi11lte
(Abohan) groups WIth respect to ordlllary multIplIcation: (a) {2 k } (lc"" 0, ± I, ± 2, .. ). I +2nL) (b) { 1 +2n J (m, n""O, ±l, ±2, .. ). (0) {cos 0 +i sm O}, where 0 IS a ratIOnal number.
(2) vVhy do the pOSltlVe rational numbers not form a group when the law of compositIOn for ct and b be definfJcl as alb? (3) If A be the operation whICh replaces tV by 0:;1; + {3, provo
a
26
INTRODUCTiON TO THE THEORY OF FINITE GROUPS
that A is of finite order If, and only If, rl.IS a root of HUltyothor than 1 In examples (4) to (8) the associatIve la,w iB assumed to hold for all elements lllvolved (4) If A, Band AB are of order 2, prove that A and B
commute (5) Prove that the elements
AB and BA have the same
order (6) If BA == A mB", prove that the
elements A mBn-2, and AB-l have the same order. (7) If B-IAB=Ak, prove that B-rA8Br=A8!,r. (8) If AB=o'BAk, show that A"B"=BvA"k" and (BvA")! = BlvAw, where w=u(7ctu -l)/Uc v -1). (9) Prove that a group IS Abelian if, and only 1l', t,he correspondence A+-+A -1, B+->-B-l, ... repre:;ents an ISOmorphism. (10) Show that a group of even order conta1llS an odd' number of eloments of order 2. (11) Show that tho matrices
Am-~B"
[0w wJ [0 w2J 0, °, W
2
whem = 1, w=I= 1, form a group of ardor' 6 with roslJect to matrix multlplrcatl0n Provo that tIllS group is Isomorplnc WIth that disoussod 011 p. 12 (12) Show -that the identioal operat,lOn and tho rotatIOns through 'Tf about any ana of three mutually perpendICular intersecting lines Iorm a group of order 4. Oonstruot Its multiplicatIOn table. (13) Fmd the order of each eloment in the llluitiphcatlve group of residues 1, 2, 3, 4, 5, 6 prIme to 7. Show that the group IS cychc of order 6, and that It ean be generated by 3 alld 5, but not by any of the other elelllonts Write 5 as a produet of two reSIdues of orders 2 and 3 respectively. (14) Prove that if a finite set of matricel> form a group, the latent roots of eaoh matmr are roots of unity. (15) Show that the set of all matrIces w3
A(V)=(l-~rl [:aV ~Vl,
THE GROUP CONCEPT
27
where v varies in the interval - c < v < c, c being a posItive constant, form a " contmUOllS " group In the sense that A(v,)A(v,) ==A(v,),
where ANSWERS.
(Lorentz Group)
12. Table 6, p. 21.
13. 5 == 6.2 (mod. 7).
CIIAJ?TER
II
COMPLEXES AND SUBGROUPS 10. The Ca.lculus of Complexes. In this chavter we shall discuss some general propertIes of abstract (.tillitC) groups. "Ve imagmc that a certam group (2.1)
of order (/ is given and that all elements wIth which we are concerned belong to thIS group In particular it wIll be assumed that the assocwtwe law and the cnncelling ntle (p. 9) have been established once for all We shall find It convement to examme subsets or complexes of elements of G In order to exprm;s that a complex K consists of the elements A, B, 0, .. wo use the notation (2.2)
We would emphasIze that thIS symbol K does not represent an element of G, but a collectIon of such elements 'rho only law of composition for group elemcnts continues to be denoted by the conventional formalism of multiplication, and no element can be regarded as the sum of two elements, to put it more briefly, the" + " SIgns m (22) stand for the word " and", and not for " pIns". A complex is conslClered completely given if the distinct clements occurring in It are known, no account bemg taken of theIr order or of duphcates among them; thus
A+B+A +O+B+A=A +B+O=B+A +0= ... We shall learn to manipulate complexes as If they were independent entIties, though different m nature from group 28
COMPLEXES AND SUBGROUPS
29
elementR, and we shall establiRh two laws of composition for complexes, namcly, addition and mult'"lJlwation. The SUlU of two complexes K and L is the complex which cOlmists of all elements of K and L combmcd. Thus if
K=J(1+K 2+ +K~ and L=L1 +£2+' +£1, (2.3) K+L=K 1+ .. +Kx,+L 1+ .. +L1 (2.4) E g , If K = it + B
+ C I1nd L = A + B + D we luwe K+L=A+B+O+D.
Addition of cornplexcR is obviously conwmtatwe and ussomatwe, i.e.
K+L=L+K,
(K+L) +M=K+(L+M).
Note that
K+K==K,
(25)
since duplicates of elements are ignored The product of two complexes is the complex obtained by formal expanmon, thus
KL=(J(1+K 2 +
+1(,.)(L1 +L2 +... +Ll) = KILl +K 1L 2 +.. +KI.Ll
(26)
E.g, in the group l'eprcsented by Table 1, p 12 we have (1 +A +D)(B+D)=B+D+AB+AD+DB+D2
=B -I-D+J +0 +0 +1 =1 +B+O +D. If Ol1e of the complexes consists of a single element P, we get and
PK =P(K1 +. . +](1.) =P K 1 +... +Pl{k' MultiplIcation of complexes is in general non-commutativo, but It IS associative and distributive, thus
30
INTRODUCTION TO THE THEORY OF FINITE GROUPS
K(LM) = (KL)M, K(L+M) =KL+KM, (K + L)M =KM + LM.
(2 7) (2,8)
(2 9)
These properties follow at once from the definitlOns. If each element of K be an element of L, we say K IS contained in L, and we WrIte Kc L or
L:JK;
this lllcludes the case in whICh K and L are Eviclently, the two relatlOns KcL
III
fact identical.
and K:JL
hold sImultaneously if, and only If, K=L It IS, m general, not permissible to apply the cancellmg rule to complexes, I.C. from the equation KL=KM It does not follow that L=M
. (2.10)
However, If one factor be a single element P, we can m£er (2.10) whenever PL=PM
or LP=MP
. (211)
holds, because (211) may be mulLlpliecl by p-I on the left or on the right, yleldmg (210) on account of (2.7) Note that 21 XcK, then PXcPK and XQcKQ where P and Q are any elements whatever When, in exceptIOnal cases, two complexes do commute we write ' . (212) KL=LK. This does not mean that every element of K commutes with every element of L; all that is Implied by (212) is that every element of KL is some element of LK, and mce versa, 1.e, that every product of the form K rx Lf3 can also be written
COMPLEXES AND SUBGROUPS
31
in the form LI'K" and that every element of the form LyKs is equal to an element K>..Lp. In partICular, If a complex K commutes with a single element P we have KP=PK, which may also be WrItten P-IKP=K.
Such cases will be of great ImporLance later (Chapter IV). 11. Subgroups. WeareparLlCula,rlyinterestecl in those complexes of G whose elementH ohey the group postulates, Huch complexes are called SUbgroups ot G Evcl'y group G h[1,s two trivial or improper SUbgroups, namely, G Itself awl the group which comnsts of the lmit clement hy Itself (1 2 = 1), all other subgroup:; alO called proper subgroups. In order that H=(Hl l H 2 , . , Hit) be a subgroup of G, its elements must satisfy the fundamental group postulates contailleu III the definitlOn OIl P 2 or, III the caHe of finite groups, the equivalent reqUlrements lai
32
INTRODUCTION TO THE THEORY OF FINITE GROUPS
CRI'l'ERION 1. A complex H of fF oup 1f, and only if, H 2 cH.
(t
fimte gronp G ~8 a su/;. . (213)
Proof (i) 'rhe relatIOn (2 13) means that the product of any two elements of H IS contameu III H, 1 e that H is closed; hence If (2 13) is fulfilled, H is a subgroup. (ii) Conversely, if H IS a subgroup, it IS closed and therefore (2.13) IS true. It will presently be shown that (2 13) may be replaced by a more preCIse statement, but we shall first estabhsh the following LEMMA
I
If H be a group and
~J
H be anyone of 1tS
elements, then
HH=H=HH Proof
(214)
Since His closeu we have
HHcH, where H is any element of H complexes
H=H1+H z +.
. (2.15)
On the other hand, the
+H" and HH=H1H+HzH+ ... +H/.H
contain the same number of elements, because we inIer from the cancelling rule that H1H,* HjH unless HI = HJ • Hence we may modify (2 J5) and state that HH =H. Similarly, we can show that HH = H.
COROLLARY. If H be a group and if C be a complex contctined in H, then . (2.16) HC=H=CH. . For let C = H' + H" + ... ; then by the lemma
HC=HH'+HH"+
=H+H+ .. =H
on discarding duplIcates (see (2.5)). On applying this result to the case III which C = H we see that, if H be a group, HZ = H. Conversely, if thIS equatIOn be satisfied, we have a fortwr~ HZ c H whence we
COMPLEXES AND SUBGROUPS
deduce from CrIterIOn 1 (p 32) that H IS Thus we have esta,bhshed CRITERION
2
gloup if, and only
In
33
fact a group.
A complex H of a fimte group G is a 8ub~f,
. (2 17)
From a given subgroup it IS often possible to derive further subgroups by meanA of the following theorem THEOREM 2. If H be a 8ubgroup and if I) be any element of G, then the complex ~8
a subgl'ou]J 180morphw with H, though not nrrrSS(l/ il!J from It. Proof Using Criterion 2 we ha,ve by hypothesiH 1-/2 = H.
di8t~nct
Hence (H')2 = (P-1HP) (P-1HP) =P-1H2p =1'-1H]> =H',
i.e H' is a subgroup. Let
H=H1 +H2+ . . +H"
and consIder the (1, l)-correspomlence H,'
~P-1H,P
between the elements of H and H'. form
An equatIOn of the
then implies that P-1(H,HJ )P =P-IJhP, (P-IH,P) (P-lH,P) =P-1lhP,
11,'11,' =11,,', which proves that the two groups are isomorphic (p. 19). 12. Lagrange's TheOl'eIn. We begl11 by a more detaIled study of the complexes HX, where H It! a Hubgroup but X is now any element of G, not necessalily contained in H.
34
INTRODUCTION TO THE THEORY OF FINITE GROUPS
LEMMA 2. If H be a sub(JrQup and if I{, and S be any two elements oj G, then the com plAxes HI{, antl HS are identzcal zj, and only if, RS-l c H; othenmse they have no element zn common. S1m~larly, the complexes RH and SH a,re identlCal1f, and only ~f, S-lRc H, otherwise they hame no element ~n common. Proof. Let RS-l =H, where He H. Then by Lemma 1,
H(ES-l)
=
(HR)S-l =H,
whence on multlplymg by S, HR =HS
. (2.18)
Conversely, let us suppose that (2 18) is fulfilled, i.e that every element of Hit is an element of HS, and vice versa; If H=I+H 2 + . . +H,.,.
(219)
a typical clement of HR IS H,R, anyone of these elemAntfl, mcluding IE, must be of the form HIS, thus
IR=HIS, R8- 1 =H, c H.
Again, If HE and HS have an element in common, we have an equatIOn of the form HLR=HJS,
whence
RS-l =H, -IHj,
where the l'lght-hanu sIde represents an element of H because H is a group (a closed set). It then follows from the first part of this proof that the complexes (2 18) are in fact identical. We are now m a position to prove one of the oldest and most important theorems on groups. THEOREM 3 (Lagrange) If H is a subgroup of G where the orders oj these groups are hand g respectively, then h is a factor of g, i.e. (220) g=nh
COMPLEXES AND SUBGROUPS
The mteger n ~s called the index oj H in G. Thete exists a set of n elements R I , R 2 , that G=HRI +-HR 2 +-
and a set 81>
82,
•• ,
•
35
,R n tn G such
. +- HR"
. (221)
• +-SnH.
. (2.22)
Sn l:Juch that
G =SIH -I-8 2H -I-
The eq1latlons (2.21) and (2.22) are respectively t'eferred to as the decompos'ltions of G into right-hand or left-hand cosets * relatwe to H. Proof. The clements R I, R 2, •• , B n will be determined one by one Let R] he any element of H, e g , I:\IllCe H is a group, we may put ill =1 We then have HRI=H If H,* G, there OXISts an element B 2 at G which is not contained in H The complexes Hand HR 2 cannot he Identical because one of the elements of HR 2 IS lR 2 , i e. R 2 , which IS not an element of H Thus, by Lemma 2, the complexes Hand HR 2 have no element in common Hence the complex HR I +- HR 2 • (2.23)
contains 2h dlstmct elements of G If there be an element R a of G whlCh IS not cOlltamed III (2.23), then HR a, whICh Includes R a among its elemelltR, differs from Hand HR 2 and therefore has no element In Gallman with either of these complexes Hence HRc l-HR 2 +HR a
. (2.24)
consists of 3h elements of G If this accounts for all the elements of G we have establIshed the decompositIOn reqUIred (n = 3), In the contrary case, there exists an element R 4 of G which is not contamed in (224). We then infer that the whole complex HR;! consists in fact of new elements. Thus at every stage of the proof we '" SometImes referred to ail associate complexes.
36
INTRODUCTION TO THE THEORY OF FINITE GROUPS
discover that either the group G has been exhausted or that at least h elements arc stIll left Smce G lA finite, thIS process must come to an end after n steps, f:lay; and we have arrived at the result that G=HR l +HR 2 + .. +HR n,
(2.25)
where each of the complexes on the right contains h elements and no two complcxes have an element III common. Thus on comparing the number of elements on both sides of (2.25) we find that g=h+h+. +h, g=hn The elements R, are not uniquely determined, for if H be any element of H whatsoever, HR. = (HH)R.=H(HR;),
so that, for our purposes, R. may be replaced by HR•. On the other hand, the aggregate of all dlAtinct complexes of the form HX, wherc X is any element of G, is of course completely determined by the groups G and H It IS true that we can formally Wl'lte down g such complexes, namcly, (2.26) where However, if Ga=HG[l,
where H is anyone of the h elements of H, we have HGa=HG[l,
so that each complex in (2.26) IS repeated h tImes and only glh (=n) of them arc distinct The n complexes HR l ,
HR 2 ,
.,
HR"".
. (227)
which occur on thc right-hand side of (2.25) are actually distinct, and it is for thIS reason that (2.27) is called a
37
COMPLEXES AND:SUBGROUPS
complete S!Jstem of right-hand eosets of G relative to H, it includes as Its first member the group H itself. The remaining complexes are not groups because they do not contam the element 1. Similar remarks apply to the left-hand cosets YH. By repeating the arguments which led to (225) we arrive at a decompositIOn of the form G =81 H +8 2H +
.. +SnH
. (2.28)
yielclmg, as before, the equatIOn g =hn.
One of the terms on the right-hand side of (228) is equal to H, say Sl H=H,
where 8 1 i/:l (Lny element of H, e g , 8 1 =1. It should be noted that not only the order 10 of the subgroup H but also its index n( =gjh) is a factor of g. E.oample In the group of order 6
G 1, A, B, 0, D, E given in Table] (p 12) the complex
H=1+0 forms a subgroup of order 2 smra 0 2 =1, or, if we wish to use Oriterion 2 (p. 33), because H2= (1 +0)(1 +0) =1 +0 +0 +0 2=1 +0 +0 +1 =H+H=H. The index of H m G IS 3 (=6...:...2) and the elements Rl' R 2 , R s in (2 25) may be taken as 1, A, B respectIvely, in fact G=H +HA +HB = (l +0) + (1 +O)A + (1 +C)B
=1 +C+A +D+B+E. We shall now deuve some simple consequences of Lagrange's theorem
38
INTRODUCTION TO THE THeORY OF FINITE GROUPS
OOROLLARY 1. If G be a group of order g, the order of every element of G ~s a factor of g P100]. Let A be an element of G of order a; then G contams the elements
I,A,A2,
. .,Aa-l,
(Aa=I)
whlCh form a cyclic subgroup of order a Hence, by Lagrange's theorem, a IS a factor of g. Example In the group of Table 1 (p. 12) the elements can only be of orders 1, 2, 3 or 6. In fact, It IS eaSIly verified that the orders of I, A, B, 0, D, E are 1, 3, 3, 2, 2, 2 respectively, OOROLLARY 2. A g10Up of pnme order has no proper subgroups and 1,8 necessanly cyclic. Proof. If the order of the group be a prime number p, the order of a subgroup must be eIther 1 or p, i.e. the sub· group conSIsts eIther of the single element I or contams all p elements of the group. If A be all element other than 1, its order, being greater than 1, is necessarily equal to p. Hence the p elements I, A, A2, . , ., Ap-I (AP =1) are all the elements of G in S3me order.
13. SUbgroups of a Cyclic Group.
G=I+A+ ... +Aq-l,
A cyclic group
(Aa=I)
whlCh is generated by the element A, will simply be denoted by G={A}. Wc can obtain complete information about all its pOSSIble subgroups from THEOREM 4. All subgroups of a cyclUJ group are cyclic. If {A} be a cyclic group of order g, then cDr'respond'mg to every d~visor h of g there ex~sts one, and only one, s1tbgroup of order h, which may be generated by Au/h.
39
COMPLEXES AND SUBGROUPS
Proof.
(i) Let g=hn
The elements
I, An, A 2n, ... , A
. (229)
are distinct, since an equality between any two of them would lead to a relation Aln=I,
where
o
This would contrachct the assumptlOn that A is of order g. The elements (2 29) form a cychc subgroup {An}
of order h, as required (ii) Conversely, suppose that
H=I+A 1 +A 2 + ... +A h- 1 is a subgroup of order h of {A}. {A}, it must be of the form
•
(230)
Smce A, is an element of
At=At. I ,
where A, is a certain mteger such that O
As H IS of order h, the h1h power of each of Its elements IS equal to the unit element. Thus Al'=Aht.I=I.
It follows that hAt is a multiple of g (Chapter I, Theorem 3, p. 22), say whence Hence At =At.1= (An)kl ,
which shows that each element of (2 30) is m fact a power of An. But we have seen that not more than h of these
40
INTRODUCTION TO THE THEORY OF FINITE GROUPS
powers are chstinct, namely the h clements lIsterl in (229). Hence the h elements of H in (2 30) are the same as those in (229). Thus
H = {A?2}, which proves the theorem. We have seen (Corollary 2, p. 38) that groups of prime ordcr have no proper subgroups Tho converse of thIS propositIOn IS also true, namely THEOREM 5. Every fin~te (lro'up of composite order has lJ1'oper subgroups. P7'oof When the group is not cyclic, no flmgle element generates the whole group, hence If A be any element other than 1, then {A} is a proper subgroup, In the case of cyclic groups of composite order the eXIstence of proper subgroups is safeguarded by Theorem 4.
14. The H.C.F. and the L.C.M. The elements that are
common to a number of complexes
K, L,
(2.31)
of G form a complex
D = (K, L, .) •
. (2.32)
whICh is called the intersection of K, L, ... When the intersection IS empty, i.e. when the complexes (2.31) have no clement in common we write
O=(K,L, ...). It is easy to see that, if P and Q be any elements of G, then PDQ = (PKQ, PLQ, ...) . (233)
where PDQ IS to be interpret,cd a,s zero if D is zero. In order to prove (2 33) let
(PKQ, PLQ, ...) =D'.
COMPLEXES AND SUBGROUPS
41
If Z' be any element of D'. 1.e If
Z'cPKQ,
we have
Z'ePLQ,.
P-IZ'Q- 1cK, P-IZ'Q-1eL, .. ,
which means that P-IZ'Q-l belongH to D; thus we have shown that P-ID'Q-1c D, or (;;ee p. 30) D'ePDQ.
. (2.34)
In particular, if D is empty so II'! D'. Conversely, if Z be t~ny elelllmlt of D, ii, {allows that PZQ is an element of D', so tlmt l'DQcD'.
The two relations that
(2.~H)
. (235)
and (2 :15) together nnply (see p. 30)
PDQ=D', winch proves (2.33) We note that, m particular, P-1DP = (P-IKP, P-ILP, ...).
(2.3G)
When the complexes K, L, .. are subgroups, the11' inter. sectlOn is never empty, as all subgroups have at least the unit element in common In suoh cases the intersectlOn is m many respects analogous to the highest common factor (H.C.F.) of mtegers and IS often call~d by that name THJ~OREM 6 The H.C F. (intersectIOn) of subgroups A, B, C, .. is a subgroup (proper or improper) of each of the (Jmups A, B, C, . Proof. Let D = (A, B, C, ..). By definition we have
DcA, DeB, DeC" .. and It only remains for us to prove that D is a group, Le. that it IS closed If X and Y belong to D, we have XcA,
XeB,
XeC,
YeA,
YeB,
YeC".
.,
and D
42
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Since A IS a group It follows that X YeA, and sImilarly that XYcB, XYcC, " . Hence XYcD,
wInch means that D ii'l closed Let A, B, .. , F be any elements of G and consider all possible products consIstmg of a fimte nnmber of factors selected from these elements with or without repetitIons, e.g BAFA2B. The aggregate of all products obtained in thIS way is denoted by
M={A, B, ..., F} and is called the least oommon multtple (L.G.M.) of A, B, ..., F From the definitIOn It follows at once that M is closed because on multIplying two such products we get another product of a finite number of factors. Every element of M is of course cantamed III G Thus we sec that M is a fmbgroup of G; m fact, It is the smallest subgroup whIch includes A, B, ..., F (~mong its elements, smce any group whwh contains these elements necessarily contams all their products and therefore 1Uts M a.~ a subgroup. More generally we mu.y define the L C M of u. set of complexes M = {K, L, ...} . (2 37) as the totu.hty of products of a .finite number of elements chosen in all possible ways from the elements of K, L, ... If
K=K 1 +K 2 + ., L=L1 +L 2 +" , and therefore
K+L=K1 +K 2 +·· +L1 +L 2 +···, we may write
{K,L, . }={K1 ,J[2'"
,Ll>L2 ,
.•
}={K+L+ .. .}. (2.38)
A typical element of M may be expressed as X =A 1A 2 •• • As>
43
COMPLEXES AND SUBGROUPS
I,et P be any
where the A's are elements of K, L, element of G and oonsidcr the group M' = {P-IKP, P-ILP,
..}
(2 3\)
Smce
P-IXP = (P-IAIP)(P-IA~) .. (P-IA.P), wc infer that P-IXP belongs to M' , as it if! a product of elements from P-IKP, p-lLP, . Thus we have shown that P-1MPcM' • (204.0) On reversing the r61es of M and M' whom P iH to be rephtC'oJ hy P-l, we arnvc at, the result that PM'P-IcM,
whICh together WIth (2.40) implies that P-lMP = {P-IKP, p-1LP, .}
. (241)
If
G ={A, B, ..., F} the elements A, B, .. ,]I' are said to generate the group G Thus the'group of order 6, whose abstract multiplication table was gIven on p 12, can be generated by the two elements A and C, III terms of which eaoh of the SIX elements can be expressed, namely
I=02(=A3), A=A, B=A2, 0=0, D=OA, E=AO, We may therefore denote that group sImply by
{A, C}. However, it must not be supposed that the set of generators is uniquely determmed, for example, in the present case we have
{A, C'}={B, D}, since A and 0 may in turn be expressed in terms of B ancI D, namely
44
INTRODUCTION TO THE THEORY OF FINITE GROUPS
It is obvious that every finite group can he generated by a fimte number of elements. We say that the elements . (242) are mdependent ~f none of them can be expressed in terms of the others, M ~f Gt is not an element of {GI , G2 , .• , G'-l' G'+l' ..., Gm}. A system of generators IS any set of elements which generate tho group. SUPPOflO that
G={G I , O2 , •• , Om}' • (243) We may evidently omit any generator that depends on the others without changing the resulting group (2.43) In fact we can always reduce a gIven set of generators untIl they become independent. Hence every finite group possesses at least one set of independent genemtors. The mdependence of the generating elements does not preclude the eXIstence of eertam relatIOns betweon thorn Thus, returning to our previous example, we can easily verify from the multiplIcation table (p. 12) that AC=CA2,
01'
(244)
(AC)2=I.
Nevertheless, the elements A and C are independent, as it is clearly impossible to expI'ess A m terms of C, or vICe versa. An equatIOn like (2.44) is called a defining relation It is often convenient to specify a group by a set oj ~ndependent genemtors, for which the assoc~at~ve law is ass1tmed, together with a system of defin~ng 1 elatLons sufficient to construct the complete mttltiplication table. Thus the group of Table 1 may he given by A3=C2=(AC)2=I, . (2.45) where the first two relatIOns express that A and C are of orders 3 and 2 respectively Using only Lhe information contained in (245) and the associative law, we might argue as follows' the six elements 1, A, A 2, C, CA, CA 2
•
•
(2.46)
are certainly distinct, as any equality between them would
COMPLEXES AND SUBGROUPS
45
immedIately result in a contradiction, e.g., If A were equal to OA2, it would follow that A-t=O, which IS incompatible wIth the assumption that A and 0 are independent. On the other hand, the elements (246) form a closed Ret In VIrtue of (2.45) , e.g., (OA)(OA 2) =O(AO)A 2=00A 2A2=02A
A A2 0 OA OA2
I
A
.112
I A A2
A A2
.1"12 1
C
OA
-
0
OLl
OA2
0 OA2 OA
OA 0 OA2 A I
OA2 OA 0 A2 A
A2
I
--- -- - - - - - - - - -
A OA2 OA 0.11 2 0 OA2 C OA I
I .112
A
Table 7
The reader Wlll have no difficulty in convincing himself that tlus table is IdentlCal WIth Table 1 (p. 12), prOVIded the elements (246) are renamed
I, A, B, 0, D, E respectively. 15. The Direct Product. From any two abstract groups G and H of orders (J and h respectively, a new group of order gh may be derived, which is denoted by Gx H or HxG and is called the dlrect product of G and H. Its elements are all pOSSIble pairs of elements, one from G and one from H, and may be written (0, B) or GxH or GB,
46
INTRODUCTION TO THE THEORY OF FINITE GROUPS
where (} and H are typical clements of G and H reHpectIvely As regards the law of composItion It IS stIpnlltted that every element of G shall commutc with: every element of H, [Lnd that the elements of G [Lnd H sh[Lll be completely mde· pendent. Thus we have . (2 t!7)
This rule automatlCally est[Lbhshes the assoczatwe law for the derived group. The 'llmt element of G x H IS the product of the unit elements of G and H When both G and H are subgroups of a group F, theIr direct product is isomorphIc wIth {G, H} If, and only if,
(i) every element oj G commutes w~th every element oj H, (Ii) G and H have only the element I 2n common It follows that the elements of G and H are then mdepelldent in the sense that an equation of the form GH =1,
i.e
G=H-l,
always implIes that G = 1 and H = I We see at once from (247) th[Lt the (Urect p1Od2Wt of Abel~an groups ~s Abelzan. More generally we C[Ln definc the chrect product or k groups, it is denoted by
G1 x G2 X
•••
x G" ;
a tY]1lC'al element m[Ly be written Ie
IT Gil,
. (2.48)
11=]
where G>. IS a typICal element of G>. The f[Lctors in (2.48) commute with one another and are independent 111 the sense that Ie
II 0>..=1 11"'1
if, and only if, each Gil is equal to the umt element
COMPLEXES AND SUBGROUPS
47
The least positIve residues prime to the modulus 15 are 1,2,4,7,8,11,13, 14. . . (2.40) They form an Abelian group of order 8 (pp 15.18) which, as we shall now r,how, is equivalent to the direct product of two cyclic groups generated by the elements 2 and 11 respectively In facL the residue 2 generaLes a cychc group of order 4, namely Example.
1,2,'i,R, 2,I=I()=I (mod 15) SImIlarly, 11 gonceates tt C'yclic gronp of order 2, (4:
(2
1, 11; 112 = 121 "" 1 (mac! Iii)
The element,s of (1 x C2 ttre obtained by InllltIplying the elements of C4 by those of C2 , thus C4 x
(2'
1, 2,
+,
H, 11, 22, 4-!, 8t1.
On bemg reduced to the Iml,Ht positIve reSIdue relatIve to 15, these numbers become 1,2,4,8,11,7,14, 13, whic11 agrees WIth t2..i'i)). The gwup IS therefore isomorphIC with (1 x (2 The followmg lemma will be usee} m the next section, but IS also of some intrinsic interest LEMMA 3 If each element of G, other than the ~,nit element. be of order 2, then G is Abelian ([nil ~som011)hlC 1J)~th
C2 XC 2 X •.• X(2 Its order is a power of 2. Proof. The proposition is ObvIOllRly true when Gis thc (only) group of ordcr 2. Suppose then that the order of G is greater than 2 and let A and B be any two distinct
elements other than I We have /[2=1, ie. A=A-l and
48
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Next consIder the element AB. Evidently AB*I, or else A =B-I =B. Hence, by hypothesIs, AB is of order 2, i e J = (AB)2=A(BA)B,
BA =A-IB-I=AB, whIch proves that G IS Abelian. LIke every finite group G posr,esse1'l a finite set of mdependent generators, say
All A z, , , "
A~.
Smce the group is AbelIan, powers of the same element may be collected m any product. Thus the general element of G Call be WrItten
G = A {J:IA2<X' ...
Akot~,
where O(t=O or 1 ('i=l, 2, ... , k), hecause the square of each element IS equal to I Hence G=
{AI} x {A 2}
X •••
x {A.,},
and the order of Gis fl=2x2x ... x2=2". 16. Survey of Groups up to Order 8. No successful method has yet been dIscovered for constructing all possible abstract groups of preassIgned order, nor do we know in advance how many such groups exist, excepL in a few simple cases. The modest means with which we have so far furnished the reader will, however, suffice to gIve a complete list of groups up to order 8. We shall speCIfy each of these groups by a set of defining relations. Since groups of prime order have already been dIscussed (Corollary 2, p 38), it only remains for us to consider in more detaIl the cases in which fl=4
or 6 or
8.
There are two groups of order 4, both of them Abelian
49
COMPLEXES AND SUBGROUPS
For If g = +, any element, other than 1, can only be of order 4 or 2 (Corollary 1, 11 3R). (1) If G contains an element A of order 4, this element generates G; 1ll fact the four elementl'l of G can be wntten . (2.50) and we have G = C4 , a eyche b'l'OUP of order 4 (2) Next, suppose that every element, exeepL 1, order 2 Hence by Lemma 3 (p. 47)
If!
of
G=C 2 x C2, i.e. G is generated by two elemenis .A and B, Etnd the four elements of G can be wrlLten
1, A, B, AB, .
(2 GI)
where
A2=B2=1, AB=BA.
(2,fi2}
This group is raIled the four-group or quadratlc group (F Klein's" Vwrer(j1 uplJe "); it is often denoter! by V There being nO other possibIlities, we conclude that any grmtp Df Drder 4 1S iSDnlDrphir, mUler luth C4 Dr IVlth V We have already encountered both these gronps in Chapter I In fact, Table 5 (p 20) represents C4 since B =.A 2, 0 =A 3, while Table 6 (p. 21) corresponds to V, provided we write C=AB 'Phere are tWD groups Df Drder 6, one eyehe and one non· Abehan. (1) If G possesses an element A of order 6, then
G={A}=C a• (2) Next, snppose there 11'1 no element of order 6; the order of every clement, other than I, IS therefore either 2 or 3 (Corollary 1, p. 38). Since the order of G ill not a power of 2, not all its elements can he of order 2 (I,emma 3, p 47). Hence there exists at least one element A of order 3, so that . (2.53)
50
INTRODUCTION TO THE THEORY OF FINITE GROUPS
are till ee distmct clements of G. If B be a further clement, it Il> easy to see that the 8ix elements
I, .A, A 2, B, AB, LPB .
. (2.54)
arc chstmct, because an equalIty hetween any two of them would ImmedIately lead to a contradICtIOn, e g. A 2B = A would imply t,hat B=A-l=A2, whereas we had !:mpposerl that B was not one of the elements (253) If the set (254) IS to form a group of order 6, the couditIOn of closure must be fulf).lled In partICulm, B2 must be equal to one of the elements (25'.1:), as we cannot have an equatIOn of the form B2 =A'B (it would be incompatible with the independence of A and B), there are only the following three possibilities (a) B2=1,
(b) B2=A,
(e) B2=A2
. (2.55)
In the last two cases B would be of order 3, Le. B3 = I ; but on post-multiplymg (b) or (c) by B we should get I =AB or 1 =A 2B, neither of whICh can pos~ubly be true Thuli it follows that (a) must hold, i e
B2 =1.
. (2.56)
Next conSIder the element BA; It must be contained in (254) As It cannot be equal to B or to a power of A, we are left with the alternatIVes (a) BA =AB,
(b) BA =A2B. .
(2.57)
In the first case G wonl<.l be Abehan, in which caRe
(AB)2 =A 2B2 =.A 2* I,
(AB)3 =A 3Bs =B* I,
and the element AB would be of Older 6, contrary to the assumptlOn made at the begmmng of thIS sectlOn. Thus we arc forced to conclude that
BA=A2B,
Ie (AB)2=I.
Subject to the associative law being confirmed, the relations . (2.58)
COMPLEXES AND SUBGROUPS
5\
completely define the group whicll we first llltroduced in Table 1 amI have repeatedly lhseussed ,unce, especially in Table 7 (see (245), p. 45). There are no other groups of order 6 'l.'he1·e are five groups oj ordM' 8 oj which tMee are Abel~an and two are non·Abehan Three AbelIan groups of order 8 are eaf:llly wrItten down, namely' (1) (8 = {A}, where AS =1 (Table 8, p. 54-). (2) (~x Cz={A, B}, where A4=B2= I, AB=BA
(Table fl, p 0--1:). (:3) Cz x Cz X C2 = {A, B, C}, where A 2 =B2 =0 2=1, AB=BA,BC=OB,UA =AU (Table lO,p. 55)
In searching for further groups of oruer 8 we lIlay assulIle that the maXImum order of elements 18 less than 8 and greater than 2 (Lemma 3, p 47). Henee there is at least one element A of order 4, I.e. • (259)
and any element other than I IS either of oruer 2 or 4. If B be an element not contallled in {A}, we ean write the. e.ight elements of the group in the form
I, A, A2, .113, B, AB, A2B, A3B..
. (2.60)
The condItion of closure demands that BZ be one of these elements, which III fact must be one of the first four, since B is independent of A. But the equations
B2=A
or B2=A3
are impossIble, as theywQuldlllply that the order of B was neither 2 nor 4 Thus there remain only two possibilities, namely (4) B2 =1 or (5) B2 =A 2 (4) Assume that BZ =1. The pJ'oduet BA must be equal to one of the last three elements of (2.60).
52
INTRODUCTION TO THE THEORY OF FINITE GROUPS
(a) If BA =.AB, we obtam the Abelian group which has already been noted under (2). (b) The relatlOn BA =A2B, which is eqmvalent to
B-IA2B=.A, is impossIble, as on squaring we should arl'lve at the result
A2= (B-IA2B)(B-IA2B) =B-IA4B=B-IIB=I, in contradIction to (2.59) conclm.ion that
Thus we are forced to the
(c)
The group whICh
IS
defined by the relatIOns
A4=B2= (AB)2=I
. (2.61)
IS called the dihedral group of order 8 (Table 11, p 1>5), or odic group. It belongs to a class of groups which we shall dISCUSS later (p flO), when the associative Jaw will be confirmed (5) Let B2 =A 2. In thIs case B is, Iil~e A, of order 4.. Again, BA must be equal to one of the last three elements of (2.60) (a) If BA =AB, the group is Abehan. On putting a =AB-l we seo that a is of oruer 2. Also, since B = a-lA, A and a may be taken as generating elements of the group, whICh IS therefore of the form (4 x (2 (type (2)) (b) The relation BA =A 2B, i.e. BA =B2B =B3, is impOSSIble, as it would imply that A =B2=A2, which is clearly absurd. Thus we must assume that (c) The relations
(2.62) do in fact define what is called the quaternion group (Table 12, p. 55).
COMPLEXES AND SUBGROUPS
53
In order to explam this name we remind the reader that a quatermon IS a hypercomplex mtmbe1' a o +a1t +a 2j
+ aa k,
whore tho ooeffiCIent,s ao, all a2' aa, are real numbers, and the four units I,i,j,lc (2.63) satIsfy tho relations
t 2 =j2= -1, ij = -jt =lc
(2.64-)
If we are only interested m the multiplicatIve properties of the symbols (2 (3), we must mterprct -1 as t 2 , !1nd the rules (2.64), which govern the algebra of tluaternions, become lc = ij and t 4 =I,
i 2 =J 2 , ji=i 3j..
. (2.65)
In this form they are indeed IdentICal WIth (2 (2), apart from the notation. The reader will have no difficulty in demonstrating that the quaternion group is isomorphic WIth the groups of matrICes generated by A2=
lV _~
V -~ ]
and
B 2 =[
-~
~
J.
or by
°IOOJ °1 0
000 0 -1
[OOIOJ000 1
and B 4 = -1
0 0 0
°-1 ° °
.
Tills affords an indirect verIfication of the associative law, which is known to be true for all matrices. The quaternion group is an Instance of a dicyclic group These are groups of order 4m defined by
A2m=!
, Am=(AB)2= B2
•
54
INTRODUCTION TO THE THEORY OF FINITE GROUPS
The elements of a dicyclic group may be written
, A2m-l,
I,A, ..
B,
AB, ... , A2m-lB.
The square of every element which does not belong to
{A }, is equal to B2. For since ABA=B, (A"'B)2_AooBAo:B=A",-I(ABA)A"'-lB=AOO-lBA"-lB = .11 00 - 2(.11 BA)A 0:-2B = .11 01 - 2BA ",-2 B = ... =AoBAoB=B2 To emphasize the structural dIf'ferencc of the five possIble groups of order 8 we append their complete multiplicatIOn tables: I I
I
A A .11 2 .11 2 .11 3 .11 3 A'" .11 4
.11 5 AS .11 6 .11 6
.11 7 .11 7
A A .11 2 .11 3 .11".11 5 .110
.11 7 I
Cs={A}, AS =1. AS .11 2 A'" A2 AS A'" AS .11 4 A" .11 4
.11 5
.110
.fl6
A6 .117
.11 7
.11 7
1
A
I A
.11 2
.11 5
A
I .11 2 .11 3
A" .11 5
A6 .11 7 I
AG AO A7 I A
A A2 .11 3
.11 2
.11"-
.11 5
A3 A"-
.11 7
A7 I A
A2 .11 3 .11 4
A6
AO
'l'able 8
A4=B2=I. 3 B AB I A .11 A3 I B AB I A A A I AB A2B .11 2 A2 AS A A2B A3B .11 2 I A3 B .11 3 I .11 2 A3B A B B A AB A2B A3B I .11 2 AB AB A2B A3B B A A2B A2B A3B B AB .11 2 .11 3 A3B A3B B AB A2B .11 3 I C4 x C2={A, B}, .11 2 .11 2 AS
Table 9
A2B A3B A2B A3B A3B B B AB AB A2B .11 3 .11 2 I .11 3 I A .11 2 A
COMPLEXES AND SUBGROUPS
55
C2 x C2 X C2 = {A, B, A}, A2=B2=02=].
A A
B 13 .1113
AB .110 BU ABO ] ] 0 .1113 .110 130 ABO A A J .110 13 ABO 130 0 ] 13 130 .110 A ABO 0 E AB 0 0 .110 130 ABO A 1 B .1113 AB .1113 B A ABO 1 130 .110 CJ .110 .110 ABO A 0 130 J .1113 13 BO 130 .11130 0 AU .11B 13 J A ABO ABU 130 .110 AB (J B A 1 ]
0
'fable 10
Dihedral Group Al=B2=I, A3 A 13 I .11 2 ]
A .112 .11 3 B AB A2B A3B
A .11 2
A .11 2 .11 3
.11 2 .11 3
.11 3
I
A
]
.11 3
I
A .112 B .11 313 .11 213 AB .1113 13 A3B .11213 A2B AB B A3B A3B A2B AB B I
B .1113 A2B A3B I
A .11 2 .11 3
BA =A3B. A13 .LoPE AB A2B .11 213 A3B .11 313 B B .1113 .11 3 .112 1
A .11 2
.11 3 I
A3B A3B B .1113 A2B A .11 2 .11 3
A
I
Table 11
Quaternion Group .11 4 =1, .11 2=13 2, BA=.c13B. A2 .11 3 AB A2B .A 3B I A B
.11 2 .11 3
.11 3
B AB i12B A3B 1 AB A2B A3B B A A .11 2 I A A2B A3B 13 AB .11 2 A .11 2 A3B B AB A2B I .11 3 .11 3 .11 3 I B A3B A2B AB .11 2 A B .112 I A B A3B A2B All AB AB AS A2 11 A2B A2B .1113 .1113 I B A B .11 3 .11 2 1 .11 313 A3B A2B AB I
I
A
A2 A3
Table 12
56
INTRODUCTION TO THE THEORY OF FINITE GROUPS
17. The Product Theorem.. At the beginning of tins chapter (2.(j) we defined the product of two complexes, K and L. We shall now examllle the case in whwh both these complexes are subgro'u.ps of G. It will appear that the product of two llubgroups (when both are regarded as complexes) IS not always a subgroup THEOREM 7 (Product Theorem) If A and B be subg1'OUpS of OIders a and b respectwely, tVlth an tnte1'sectwn (H,C.F.) of order ct, then the complex
C=AB contams exactly ab/d distinct elements .. ctnd C is a group If, and only if, A and B comm~tte, Proof (i) Smce A and B are groups, their intersection D IS a subgroup of eIther group (Theorem 6, p. 41), and
we can decompose B lUto cosets relative to D, thus B = DB I
+ DB 2 +. , ,+ DEIll
(2.66)
where n = bid and DB,,pDB3>
ifi,p,j,
. (267)
MultiplYlllg (266) on the left by A we obtalU
AB =ADB I =ADB 2 +, .. +ADB w Smce D is contained in A, it follows from the Corollary on p. 32 that AD = A, and therefore
AB =AB1 +AB 2 +. , . +AB w
•
(2,68)
Each complex ABt contams a distinct elements, and no two of these complexes have an clement in common; £01', II they had, we should get an equation of the form Aa.Bi = A~Bj,
(i
*' .1)
where Aa. and .A Jl are certam elements of A would follow that
Hence it • (2.69)
57
COMPLEXES AND SUBGROUPS
Smce A 1/:! closed the left-hand side of (2 69) would be an elemcnt of A, sImilarly, the rIght-band Slue would represent an element of B. Thl::l element would therefore be common to A and B, I e it would belong to tbe intersection D Hence, by Lcmma 1, we should have D(BJB,-l) =0, DBJ=DB,
m contradictIOn to (2.07) Thus we C'oncludo that tho complexes ABi in (2.68) have no clements m commCln and therefore eontam a total of
dlstmct element~ TillS provcR the first part of the theorem. On intelehangmg A anrt B we note that the complexes AB and BA (where A and B are groups) always contain the same number of clements even when AB BA. (ii) Next, suppose that the complox C=AB is m fact a group If A anu B be any elements of A and B respectively, we have .A -1 e A, B-1 e Band thcrefore
*
A-IB-IeAB.
Since AB is a group it contains the reCIprocal of A -IB-l, Le. (A -lB-I)-1 =BA cAB. Now BA may be considered a typIcal element of BA. Hence we have proved that BAeAB
On the other hand, we have Just seen that these two complexes contain the same number of distinct elements, so that BA cannot be a proper part of AB ; it therefore follows that
BA=AB.
(2.70) E
58
INTRODUCTION TO THE THEORY OF FINITE GROUPS
(iii) Conversely, if (2.70) IS true, we have C2 = (AB)(AB) = A(BA)B = A(AB)B
=A2B2=AB =C, using Criterion 2 (p. 33) for the groups A and B, and in virtue of the same Cl'lterlOn we infer that C is a group This completes the proof of the Product Theorom. NOTE: 'rhifl theorem is analogous to a well-known result of elementary arithmotIC, namely that If a and b bo any two integers who;;e H.C.F and L C.M are d and c respectIvely, then c '" ab/d.
18 DecomposItion relative to Two Subgroups. The decomposition of a group into cosets (Lagrange's Theorem, p. 34) was essentially based on the fact that two Auch cosets are elther identlCal or else lw,ve no element in common Whenevcr a set of complexefl possesses this property, the elements of G can be divided into mutually exclusivo classe:,l. The following generalization of the cOllcept of cosetfJ IS due to G. Frobemus..j< Let A and B be Rubgrollps of G (which lleed not be distinct) of orders a and b rospeotIvely, and consider com· plexes of the form APB, AQB, ... where P, Q, ... are any elcments of G. We shall prove that rl two such complexes have one clement m common, they are in fact Identical Indeed, let us suppose A1PB1=A 2QB 2 ,
where AI' A 2 are elements of A and B ll B 2 of B On premultiplymg by A and pestmultIplying by B, wc find that AA1PBIB =AA 2QB 2B
* Sitzung8bel'~chte Berlu~,
1895, pL.
1,
(2.71) pp 163·94.
COMPLEXES AND SUBGROUPS
S9
Since A and B are groups we have (Lemma I, p 32) AA 1 =AA 2 =A, B I B=B 2 B=B,
whence (271) becomcfl
APB =AQB We note that thc complex APB contains the element IPf, I.e. P. Ulling these factH, we can ootam a (leCOrnpofllJ,ion of Gin the following way: ehooHe any clement 1\ of G. ff the cOlllplex APIB lH le0l8 than G, let 1'2 be an element not eontamed m it Since P 2 Ii'! an element of AP 2B but not of APIB, these two complexeOl have no element in eOllllllOl1, and we can Hegregate from G the complex . (2.72) If there be an element P 3 not yet accounted for, the complex
AP3B consIsts entirely of new element'! and should bc added to (2.72). We llroceed m this way until the whole group G is exhausted No further elemcnts can then be found and we have an equatlOn of the form . (2.73)
We say that G has been decoJnlJosed relative to the subgroups
A and B. In contrast to the resolution into cosets, the number of elements in the various terms on the right-hand side of (273) may vary and requires closer examinatlOn. ConsIder a typical complex
APpB=K I +K 2 + .. +Kr whlCh contains, say, l dIstinct elements of G the complex
Pp-lAPpB=Pp-lJ(l +Pp-lK I +
Evidently
.+Pp-lKr
(2.7-1)
60
INTRODUCTION TO THE THEORY OF FINITE GROUPS
also contams I
di~tmct eleIllent~,
hecaufle If
Pp-l]((I. =p/.-l]({l we should find that I1nd mee versa
LeL
P p-lAPp =A p ThIs is a Imbgroup which is IsomorphIC with A (Theoreln 2, p. 33) and hence IS of order a. The left-hand sIde of (274) can accordmgly be regarded as the product of two groups, namely ApB On applying the Product Theorem (]). 56) we infer that l =ab/dp,
(2.75)
where dp Hi the order of the intersecti.on (A p, B) We collect these results in the folloWing theorem. THEOREM 8 (Frobcllms). IJ A and B (t?'1', snbgl'oups of G of orders a and b resp(jct~vely, G admits of a decomposition into mutually excluswe complexes relative to A and B, th1ls
G = AP1B +AP2B +.
. +APrB.
The complex APpB contmns ah/clp elements where cl p i.~ the order of the mterscctwn (Pp -lAPp, B). COROLLARY On counting the number of ell'ments in each term of (2 73) we obtain the relation
g=
r lab 2:([. p=l
. (2.76)
P
Examples (1) If H is a subgroup and K any com.plex of G, prove that HK= H or KH ",H implIes that K cH (2) Let D", (X, Y) and M =: {X, Y} where X and Y arc any complexes. If Z be another complex, show that (X, Y, zj", (0, Z)
and {X, Y, Z}:= {M, Z}.
COMPLEXES AND SUBGROUPS
61
(3) If A and B ho subgronpfl WhOflG order'"\ are rolatlVoly prIme, prove thai, theu' mLerseotlUn consists only of ~ho umG element (4) Prove that lf G=Hlt 1 +HR 2 + . • +HR n is a deeom· pOSlGlOn of G mto rIght-hand oosetfl relative to a subgroup H, then G=R 1-1H +B 2 -1H + .. -I R n-1H IS a decomp091tlOn mto left-hand 008et9, (5) Fmd all sllbgl'OUpR of ordor 4 of the UIlIl'dl'al group of ordor 8 (Table 11) (0) Show that the group of T
URAI'TER
(II
GROUPS OF PERMUTATIONS
19, The SymmetrIc Group P,,' The operatIOn of 10u,tTanging 1~ chstinct objects amongst th.emselves is called a permutahon of degree n, The objects will be denoted by letters or sImply by the numerals
1,2, ., n If the permutation A replaces 1 by a1> 2 by ({2' where
(3.1 ) ,.
,n by
an,
(32)
are the numbers (3.1) in some order, we shall Wl'lte
A - (] 2 .. n ) a1 a2 • ,an
(3 3)
to indicate that each number III the first row IS to be replaced by the number immediately below it in the second row. There aTe as many dIfferent permutatIOns of n objects as there arc arrangements "- of the type (3,2) From elementary Algebra It is known that tIm, number is equal to nl Since it is immaterial in what order the information about the n objects IS given, we may rcarnmge the columns of the symbol (3.3) at will; indeed It is always possible to modify the expression for A in such a way that the nrl:lt row consists of the numbers 1,2,. ,n in any preassigned
* \Ve use the word "arrangement" to denote the ,gequence of numbera (3,2), .'oserving the term" permutatwn " for the opewl1on whidl l'cplacclJ (3,1) by (3.2) tl2
63
GROUPS OF PERMUTATIONS
order, provided the sarno pairs are kept m vertIcal ali
n'
4-) = (23 21 43) (4- 2 1 3 ) 4 1 = 4- 3 2 1 = ....
1 2 3 ( 2 3 1 4-
By the ]Jl'o(hwt AB we mean the permutation obtained by first oarrying out A and then 'I- B. Ruppose that B _ ( 1 2 .. - . b1 b2
(([1
) a 2 • • an ) (3.4) b" = ('1 1":2 ••• r" ' . whore the seeoml symhul for B haR been derived from tho firRt by rearranging the columns until the firRt row l>eeomcR identMtl with the soooml row in the symbol (3.3) for A. TInR hal-! been dOlle to prepare BfaI' premultiplication by A. We now have n
n )( u1 u 2
an
c1 c 2 = (
•• ••
an ) Cn
~I ~2
:.
~n);
(35)
for a typICal number i is changed by A into a" and B changes at into c" hence the combmed effect of A and B (m thIS 8equence) IS to change i mto Ct, whICh agrees With the final symbol m (35) NotICe that when the second factor B has been prepared for multIplication by A, the product AB is found sImply by placmg the second row of B below the first row of A. E.g., 1£
1 2 34)
A =( 2 3 4 J AB = ( ;
and B =
~ ~ i )( i ~ ~ ~
( 1 2 34)
3 1 2 4 '
)= ( ~ ; : ; ).
On the other hand,
1 2 3 4 )( 3 1 2 4) (1 2 3 4 ) 4 2 3 1 = 4 2 3 1 .'
BA = ( 3 1 2 4
" Some authoul adhere to the opposite convention.
64
INTRODUCTION TO THE THEORY OF FINITE GROUPS
which shoWH that multiphcatlOll of permutatwns 1S, in general, not commutatlVe. WIth a little practiee the reader wtll become accustomed to evaluate products wIthout having to write out the 1Iltcrmediate stage of preparation Thus 111 the example we have Just cOllsIdeIed the :product BA may be found as followH: 1 is moved t.o 3 hy B, and 3 is moved to 4 by A , hellre hy eornhining these operations we :Jee that BA moves 1 to J. Next, 2 IS moved to 1 by B, and 1 is moved to 2 hy A; lIcnee 2 iH moved to 2 (is left unaltered) by EA, Imd HO on. Havmg definer! tho multiplIC[ttion of permutations, we H}W!l now prove 'fITWmEM 1. '1'110 set oJ all }Jerm7ttatwns oj n objects forllls n (Jroup P" of onlel' n l , ~t is called the symmetric group oj dr(Jl'ce n Pmof We shall vcnfy Jlostulates (1) to (1 V) of Chapter I, pp 1-2. (1) The closure of the set P" of all permutations of n objects follows Immediately flOm the defimtlOn of multiplIcation (ii) To prove the assoc~ative law let us contract the symbol for permutations and wnte
where rx is a typICal representative of the numbers 1, 2, ..., nand f3 is its image under the operation A. Similarly, we may assume that, after being smtably prepared, two further permutatwns may be represented by
We then have
65
GROUPS OF PERMUTATIONS
and A(BG)
=(~)[(~)G)] =(~)(~) =(~), (AB)e =A(BC).
i.e
(iii) The umt element of P" mutatIOll
I~(12 .. 1 2 ...
the ,. identical" per-
IS
nn)= =( aa
1 ((2
an )
1 ((2
It"
'
which le/wes cach object unchangcll. (iv) If
n ) (/ n
'
the inverse 11erlliutatlOIl may be represented by
A -1 = (' Indeed,
11l
(11 Ct 2 • "
1 2
an)
. n
contracted notation, we have
and
This concludes the proof of the theorem Suppose the objects 1, 2, ... , n h[we been separo.tcd into two mutually exclusIve sets, say
If A permutes only the IX'S amongHt themselves and if B operates only on the fi'R, thon the ordor in whICh A and Bare earl'led out is evidcntly irrelevant, since neither operation has an influence on the other. lIenee two 'P/~1·.
66
INTRODUCTION TO THE THEORY OF FINITE GROUPS
m~dations
which operate on tnutnally exclusive sets of objects always commute E.g., if
1 234 )
A = (, 2 1 3 4
and B =
( 1 2 34) 1 2 4 3
'
1234) AB =BA = ( 2 1 4 3 ' since A opemieR only on 1, 2 and B only on 3, 4 20. Circular PermutatIOns (Cycles). A permntatlOn which interehangcH the objects cyclIcally is called a circular permutation or cycle. Thus if m objectR be denoted by 1, 2, ..., til, H1LUh a permntatlOn is represented by
C=
(
1 2 . m -1 'In 23." 'In 1
(3.6)
)
If we vISualize the m objects arranged at m placeR on the rircumfercnre of a circ:le, the pel'mutatlOn G moves each object to the next place so that the last object comes to occupy the place of the first On account of their great importance for the theory of permutatIOns, cycles are usually WrItten III contracted notation thus
c= (1
H9
2 ... m),
indicating that each number in the bracket is to be replaced by its successor on the right, and the last n~b~qlw the first. Since it is immaterial with whieh num!lef' ~evbegin,
~le~a~~. m)=(2 3 = (3
,,'In
4. .
1)
'In
1118 8 T-
1 2) = (m 1
'In
-1).
(3 7)
We shall noW prove Tfn:oREl\f
2.
Every permutation can be 1tniquely resolved
into cycles which operate on m~ttually excl1lsive sets of obJ'ects Proof. Let A he a given permutation of 1, 2, ... , n Beginning with any oue of these jutegers, say 1\, let us
GROUPS OF PERMUTATIONS
67
suppose that A moves Ato .:\', It.' to A", N' to .:\''', and so all ; as there are only n numbers mvolved III A, there must be an index 1, where l-;;:;r,-;.,n such that
All) =A. Thus some part of the effeut of A is eqmvalent to the cycle L = (A A' A" ... A(I»). . (3.8) If r =n, all numbers have been u,ccounted for and we have A = L On the other hand, if r·' n, let j), be a number not contamed in L u,nd suppose that A carncs I), mto j),', I),' into j),", and so on untIl we return to j)" as wc Intuit do after at most n - l' stapH, for the images of j), and it'l snccessors are certltmly different from the A's, or elOle two numbers, one j), and one A, would have the 'lame image We h,wc therefore isolated another cycle M=
(j),
f-t' f-t"
..
j),(S»,
say If r +8 =n we have proved that A =L111 =llIL ('lee the remark at the end of § Hl, P 65). On the other hand, if 1 +8
+8 +
. + ~l = n.
(3.10)
This resolution into mutually exclusive cycles IS unique, save for the order In which the cycles occur (we have already seen that they all commute), and save for the alternatrve ways in whreh each cycle may be expressed (see 3.7), fol' it is clear that two essentiully different products of cycles correspond to dIfferent permutations. Example. ( 1 2 3 4 5 (l 7 H ) = (1 4 8 7)(2 5 3)(6). \ 4 5 2 836 1 7
68
INTRODUCTION TO THE THEORY OF FINITE GROUPS
A cycle wInch consIsts of a slllgie letter merely indicates that tillS letkr is unaltered, such cycles are often omitted, It bemg understood that any letter wInch is not exphcitly mentloncd, remaml:l unchanged'r When thIS conventioIl IS adopted, the complete interpretatIOn of a product of cycles dep('II(li:l on the degree (n) of tlw permutatioIl, I.e, on the numher of ohjects involved 'J'hlls
It, is
CHI:lY
~ ~
when n = 3,
(I 2 :1) =
(
") (1 2 ,)=
(123145 2 3 4 I)) ,Wlenn=o. I
{
),
h
to write down the IJowers of a gi ven cycle 0=(1 2 .,. m)
Smce C moves every letter to the next place on the oircumferonee of a circle, 0 2 sends every letter two places along, 0 3 three planes along, and so on. Generally, 0 7, Ieplaces x lJy X 1- k where this uumber has to be reduced to its IcaAt positlVe remltlllcler relative to m. In partIcular, It follows that 0 is of or(1er m, i e. (1 2
, (3.11)
because C,n replaces x by x +m, which is equivalent to x Thu8 a cycle of de(Jree m ~8 of o?'der m. E.g., when m = 6, we have (123450)2=(135)(24-6) (1 2 3 4- 5 B? = (1 4)(2 5)(3 I'l) (1 2 3 J 5 6)4 = (1 5 3) (2 6 4) (1 2 3 J 5 0)5 = (1 6 5 4 3 2) (1 2 345 G)6=I.
Let
P = 01C2
* TIll' Idcntwal pel'mutation I
it, !rorn totnl evanellcenee.
' Ur
•
(3.12)
is then usually wrItten (I) to save
GROUPS OF PERMUTATIONS
69
be the resolution of a given pcrmutatIOn into mutually excluSive cycles of orders , (3.13) respectively. (1.7))
Since thcse cycles commute we have (see pm=01"'02m .. 0/".
Suppose now that P is of order m so that • (3.14) and therefore
°
01 Yn = (OP3' .. Gr)-m
Now 02' 3 , ••• , 0,. do not permute thc letters wInch M'e mvolved in 0 1 Hence
°1
m
=1. It SimIlarly, we prove that 0 2111 =Oa m = .. =0/" ",,1 follows that 1n IS a multiple of eaoh of the mLegor>l (313) Thus the order of any permutation tS the L.C M of the orders of tts COmlJonent (m1ttually excluswe) cycles. A permutatIOn IS said Lo be regular if all its oycles are of the same degree. As an example we give a list of the six permutatIOns of P3 (n = 3), each resolved into oyeles : 1 = (1), A = (1 2 3), B = (1 3 2), G = (1 2), D = (1 3), JjJ = (2 3). We have A3=(1 2 3)3=1,
0 2 =(1 2)2=1,
(AO)2=(23)2=1.
Comparing these relatIOns with (245) P 44, we sec that Pa is isomorphio with tIIe abstraot group of Tn,ble 1, /:Joe p. I:? Thus the elements of that group may be interpreted or represented as permutatIOll>!, and all !clatiolls which fonow from the multiplicatIOn table have their' oounterpart in relations between the permutations of the group P3' This
70
INTRODUCTION TO THE THEORY OF FINITE GROUPS
affords an lllrlirect verification of the aSSoCIatIve law for the abstract group under consIderatIOn (p 12), becauRe this law IS known to hold for all permutatIOns, and in partICular for Ps. 21 Classes of Permutations. Two permutatIOns, A and B of degree n, are said to be SImIlar or conjugate" with respect to PI' If there eXIsts a permutation '1' In PI' such that B=T-IA1'
(3.15)
This concept lR of fundamental importance in the theory of grollpH and WIll be more fully dIscussed In the next chapter. 'l'he relaLIOll between two conjugate permutations IS best understood when they are resolved into cycles. Let
A =01°2 " Or lle a given pCl'lm1ta.tion dDcomposed into clusive cycles, and let '1'
=
(12
..
1
. (3.16) mutually ex-
n) (A) A'
I' 2' .. n'
=
be any permutation of PI' whatsoever. an expreSSlOn for T-IAT Since
We WIsh to find
T-IAT = ('1'-101'1')('1'-10 2'1') ... (T- 10,T),
(3.17)
it is sufficient to evaluate a typical factor '1'-10'1',
where
0=(a 1 a2
•••
am
)=(a'+l a,)
. (318)
is a cycle of degree m. (In the last symbol am +1 must be considered identical with a l .) Generally we can write • ThIS agreos wiLh tho tOrImnology of Chapter IV. In a dIfferent context, aome authOlR fopeak of conjugate permutl1tlOlls when we should URe tho torma I rec,procal' or 'inverse' (A. C. AItken, Determimi/Its and MatrioeB, 4th ed , p. 33)
GROUPS OF PERMUTATIONS
71
T-I=G'). If i\ does not occur m 0, then,\' IS not altered by the permutation T-IGT For T-I transforms A' mto A whIch is changed back mto N by T, the factor C havmg l'emamed palll'live. On the other hand, if i\ =a" we IHwe
at' ---»- a, (by T-I),
a, ---»- a'+1 (by 0), a, II ---»- a,~ I' (hy 1'), so that 'I'-IOT changes a.' into a, 1/' Thns we cau write T-10T = (aI' 1l 2' •• (( l"') (3.19)
On comparing (3.18) with (3.Hl) we observe Lhat 'I'-lOT is derived f1'Om 0 by applying lhrl permutation T to the letters m the bracket representmg C. TIns rule holds generally for any product of cycl~s E.g, If 12345' A = (1 3) (2 4 5) and T = ( 2 4 5 1 3 ) we have T-IAT =(2 5)(,1 1 3) replacrng, m the expression for A, 1 by 2, 3 by 5, ... as reqUired by T Let us suppose that in (316) the cycles are arranged in non-decreasmg order, lIlcludmg those of order 1, and let these orders be denoted by 1
'-;;fL.~n, •
(3.20)
where /.Lrl-/.Lz+'
+/.L,=n
(321)
Thus every permutatIOn of uegree n is assoCIated with a partltlon of n into positive integers, namely, the degrees of the cycles into which it is decomposed. Two permutations whwh correspond to the same partttion are sairl to belong to the same class .J< of Pn' It follows from (3 W) and (3.17) that '" Not to be oonfused with tho thsLmotwIl between even and odd permutatlOns (p 74 below). The word' class' WIll not he used by us In this oonnoxion
72
INTRODUCTION TO THE THEORY OF FINITE GROUPS
conjugate pernmtations bclong to the same class. Conversely, If A and B be two permutatIOns of the same class, say and B = (fA}
Hfli .. flit,)
. . . (WI' • W ILr ) , thon a permutation ']' can be found such that ••• fAIL'
B =11- I AT.
. (3.22)
Inueed 1£ wo put
(a
al'l 6
bit,· . ~t}. U ltr) fl", •.. WI' . Wit, ' the evaluation of the :product on the right-hand side of (3.22) amounts to replacing the letters in the cycles of A by the corresponding Greek lettols, a procesH which turns A into B. Thus we have proved the Important result·
'1' =
l •..
(Xl ••
1
(XI'.'
fll'
•
THEORElw 3. Two lJerrnulations ate conjugate w2th respect to P11 tf, and only tf, they belong to the same class. There are clearly as many classes of Pn as thore are different ways of partitIOning n in accordance with (3 20) and (321). It is Important to know how many permutations belong to a given class Instead of specIfying the class by a partitIOn (3 21), let us suppose that each of its members involves (Xl cycles of degree 1, cx z cycles of degree 2,
where
CX n
cycles of degree n,
lcx l +2cx z + ... +ncxn=n. The set of non-negative integers
(323)
(IX) : tX I , tX 2 , •• ') tX n . (324) oompletely detennmos the class (<x). To construot all l'ermutations of (cx) imagine a pattern of empty brackets
GROUPS OF PERMUTATIONS
(.)(.)
..
(.)(. )( .)
~.--'
~----
()
73
(3.25)
whIch correspond preCIsely to the oycles of a typICal permutatIOn A of (a) Thoro are altogether n blank sIJaces III the pattern, and these can be fiUeu with the letters 1, 2, ..., n in n' ways. However, not all thcse n l arrangements correHpond to dislmct I)ormulations. :For the f/.~ cyolel'J of tlegrce Ie (lc"", 1, 2, •.. , n) may be permuted III mty manner without causing a change in the resulting pcrIlluto,tion A. Thus tho sumo permutation lH ohtullletl . an! . . (3.2B) tImes If ltIl al'rltngementl'J ure cOlHnderecL Also, it I'Jhuuld be lCmOlubered that a cyc:le of degn'e k may be writton in k dttIerellt ..,\ ays Slllce eaeh of ItA lettcl's may hr brought to the leading position 1ll the bracket (p. (6). Hence Cl.k cycles of degree k admit of lcrf.~ eqUivalent arrangements of lettcrl:l yIeldmg the same permutation
n'
all 0. 21 .
let,. 20:,· ...
n('f~
. (3.27)
tImes Thus in olCler to obtum the exact number hrf. of cycles11l the class (IX), we must dIvide n' by (3 2B) and (327). This gives the formula nl h('f = ICC '(Xl '2rf.'(/2'I ••• n DC" ~Cl.w
. (328)
which is due to Uauchy.* 22. TranspositlOUS. A cycle of the form (ab) is called a transpositlOn; it interchanges the letters a and band leavei'J all the others unaltered. We note that (see (3.11))
(ab)2=I,
(ab)=(ab)-l=(ba).
Let Xl' X 2' ••• , X n
be independent varia})les and oonsitlel' the J)roduct of differences
* Exeraiae d'analysc ot de phySIque matMmatique, iii (1844), p. In. F
74
INTRODUCTION TO THE THEORY OF FINITE GROUPS
D..=(Xl-X2)(Xl-X3)(:tl-X,I) .. (Xl-,v n )) (x 2 - x 3)(X2 - t'l) . (x 2 - :t n ) . (x 3 - :t'4) •• (x a - xll ) . (X"_l -
(329)
:tn)
If the variable;! be HubJeeted to a permutation A, the function D.. lH formally changed mto a now function D.. A whlCh IS, huwever, identICally equal mther to D.. or to - D... We shall write . (3.30) D..1='(A)D.., where '(Ii) hi a function of thc permutatIOn A whlCh has the pruperty that (3.31) 'P) =101' -1. DEFINITION 1 rl pel nmtatlOn A 'is saLd to be even or odd accord!ilU as '(A) == 1 01' '(A) = - 1. 'The Junction' (A) ~8 called the alternating character of Pn The most Im-
portant fart ahout this funotlDll following THEOREM
4
IS
expressed
III
the
If A and B be any two penn11tatLons, . (3 32)
of two even or two odd pennutatLons ~s even whilst the product oj an even and an odd permutation i.s odd P1'00j. If wc apply the permutatlOn B to both sides of
i,e. the
prod~wt
(3.30), we get
D.. AB
=, =, (A)Cl. n
(il)' (B)Cl..
. (3,33)
On the other hand, if we let the permutation AB take the place of .ti, the identity (3.30) becomes D.. AB = '(.dB)D.., .
• (3 34)
whence we t1erluce (3 32) hy comparing (3.33) and (3.34). Evidently, if 1 is the identioal permutation, we have Cl.1 = D.., Le. . (3.35)
GROUPS OF PERMUTATIONS
Hence, on putting A
=
75
B-1 m (3 32), we find that
'(B)'(B-I) =1,
(3.36)
or in VIew of (331) (337) Also, If A and B be any two permutatlOns, ·we have by a repeated application of (332) ~(B--IAB) =~(ll-I)~(A)~(B)='(B-I)'(B)~(A),
whence by (3.30) (3.3H)
i.e. lJP,rmutatwlI8 oj thp chcwar-[er 'l'HmmBlIl fl.
Pl'Oof.
T~et
.WUIt(~ class
lull'P the
sml/(' aUtl
nalmy
LJ II tl'U1/8j!081tlOII8 are odd lJPllllutatlOns. A = (12). The function ~(2) IS obtained
from £:!,. by an mterchange of :t'1 and x 2 We see by inspectIon of (:.l.2U) that tbiH n1ll0UnLH to reversmg the SIgn of each difference in the first two rows, thus ~(I2)
= ( - 1)"-1( -
1)n-2~ = -~,
i e.
To obtnin the alternatmg l'haracter of other transpositions we use the IdentItIeS (la)=(2a)(l2)(2a)-1, (a»2),. (uu) = (lb)(la)(lb)-I, (l
(339) (3.40)
whence by (3.38) '( (au))
='( (la)) =,( (12)),
and therefore for all tranflpoi:litions. THEOJ.~El\l 6. [ijl'P1'l! lJcnnutatw1L can, 'in infinitely many wa1/8 , be eX1Jre8sed a8 a product of transposltlOnB, the numbe1'
76
INTRODUCTION TO THE THEORY OF FINITE GROUPS
oj transpositIOns m s1wh a IJrodllct being afways even or always odd aceO/dina as the (lIven JiBrmulation 1S even 01' odd. Proof We shall firOlt show that every ('y('le ean be exproOlsed u,s a producL of tranOlpoOlltlOns ; mdeod (a 1 a 2
am) = (a 1 a 2)(11 1 a~).
(a 1 an,),
(3.41)
can be vel'lfied by evaluatmg the product on the rlghthand Ridc' (£1--.-7- a 2 , ([2 --)- 1 ----7- a 1 , a 3 ----7- (11-»- a4' and so on Rince ('very p('rmutlLLlUn can be WI'ltten aR a product of ('ydoR, the firRt part of t,he theorem IS proved As the trnnspmlitwIlH III (3.41) havc a lettor in common, they do Hot, III general, commute. Also, the product iRnot changed by the lllscrtion of pa,irH of f(wtor8 such aH
D,fl
°
(ab)(nb), wInch arc eqUIvalent to the IdentIcal permutation For this reaSOll alone the factorization into tra.nspmHtions cannot be unique. Ruppose that a given peJ'llwtatlOn A IS expressed as a product of transpoHltlOns in any way whatever,
A =J\T 2 "T s
By Theorems
~
• (342)
and 5 we have s
'(A) =
n
1=1
'(Ttl = ( -1)8
Hence the number of factors in (3.42) IS even 01' odd accordmg as A IS even 01' odd. We sec frum (340) that every transposItion can be exprcStled III terll1R of thE:' Rpecial tram;positlOlls (llc), where k = 2, 3, .. , n. Usmg t,hc techmua.l term introduced In Chll.llter II, p. 43, we may enunciate THEORI~M
7
'l'he symmetric group Pn may be genm Cited
bll tbe n - I IranspOintions (12), (13), ... , (In)
. (3.43)
77
GROUPS OF PERMUTATIONS
23. The Alternating Gl'OUp An' We rC'tnrn to the dlf:ltinetion between even awl odd permntations mtl'ouucerl on p 74 rrUEOREM 8. In any group of permutations G eIther all or exactly half the pel1nutations are cven. 'I'1Ie even permutations of G form a group by themsdves Proof. If G contl1mB no odd pel'mntatlOll, there If!
nothing to prove. SUIJPOHC thai, (J If! all odd permutatioll helongmg to G so tlll1t '(0) = - 1 A~ X rUm; through the whole group, so doeH QX where (J if! ibwl] (p. a~) Hellr'e ):'(X) - LWJX).
x
.t
Un the other hand,
whence
L'(QX) =LWd)'(X) ~ - L,(X),
2.:'(X) =0. TIllS mel111S tlmt the Hum contain~ all equal number of pOHitive ~1,lld negative terms, 1 e. there ale as many even af! odd permutatiom in G. Alf!o, it is evirlent froUl Theorem 4 that the even permutatIOns of G obey the pmltulate of closure und therefore form a (proper 01' 11llproper) Aubgroup of G. Sreeml mterest is attached 1,0 the case m whICh G iH the symmetnc group P". DEFfNITION
2. The sct of all even permutatIOns of degree
n forms a rJl O~tp An of Ol'llcl' ~n!, which is called the alternating group of dC(J1 ee n
E.g., the alternating group A~ 18 of order H4 1), i.e 12, and conSIsts of the following permutations (arrangecl arcorclmg to classes of P4)
A4
:
(1) (=1) (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) (I 2 3), (J 2 4), (1 3 2), (1 3 4), (1 4 3), (2 3 -i),
(1 .j 2), (2 4 :1).
78
INTRODUCTION TO THE THEORY OF FINITE GROUPS
A permutation is even if, and only If, It can be written as the product of an even number of transposItIOns By Theorem 7 these transpOSItions may be seleoted from the set (3 4-3) and then arranged in pairs Since (li)(lj)
=
(hj)
*
(1 J, t> 1, j> 1),
we have 9. The alternattng group Au may be generated 2 tmwry cycles
THEOREM
iJy the
11 -
(l 2 3), (1 2 4),
., (I 2 n)
. (3.44)
24. CaylllY's Theorem. 11, was not untIL comparatively Il1,to tImt tho abstract cOllcept of a group was fully appreciated. The earlIer hteratnre on the subject, ll1cluding the olassical WOl ks of Cauohy, Galois and Jordan,'!' dealt exclusively with groups of permutations, although they contaln many rmmlts which apply equally wen to all fill1te groups A close study of permutation groups is, howe'\1"er, of more than historical intereflt, because we shall presently show that every finite group whatsoever IS isomorphic with a Imitable group 01 l)ermlltatiom; This important fact was discovered by A. Cayley. t Let . (3.45)
be an abstract group of oreIer g. If P be anyone of these elements, the products GIP, G?, ..
'j
GuP
. (3.4.6)
are fJ distinct elements of G and therefore represent arean'angement of (3 ·15) Thus with every element P of G we ('an assooiatE' 11 permutatIOn of degree g, namely
pi = (0] alP
*
a2
(J2P
•••
Ou )
. auP'
C.•Tordan. ']'Hl,iU des 8ub8t~tuNon8, Pal'l~ 1870. t Pial. Mag. vol. vii (4), 1854, pp. 40·47.
(3.47)
GROUPS OF PERMUTATIONS
79
The objects on whlCh thIS permutation opel'ates are the group elements themselve:;. It is convcnient to use the abbreviated notatIOn P' = (GaT',p)
<) (~=1, "'"
" g)
. (3.48)
When 0, runs through the whole group, so does a,x where X is a fixed element of G The mfol'mation contained m (3.48) may therefore also be expressecl hy
(O,X ) 1)' ... - O,XP ,
(3,40)
Rimilady, let (,I' = v
({,,'i Q\ (
f,
)
•
3 "0)
(, ••J
be the permutatIOll corresponding to the elemen1, Q. If P and Q are di:;tinct, the permutatIOns (3.41:\) and (3 50) arc certainly not equal; indeed, Rlllce OIl' * O,Q, they have a rlifferent effect on each of the permuted objects. Hence we have estabhshed a (1, I).correspondence between the elementi> P, Q, " of G and the set (348) uf permutations P',Q'" . . Next, let 11S evaluate the product P,Q, ; we find that
P'O'" _(G, )(0, ) _(G, )(G,P ) _(G, ) - (J,P ,G,Q - (),P, O,PQ - G,PQ The result is the permutation which corresponds to the grouJl element PQ , i e. we have (PQ)' =P'Q', . (3.51) Thus we havo shown that the eOl'l'cspondence P -<E---? P' is In fact an isommphisTn 'rhe set of pCl'mntations of degree (J G': P', r/, forms a gronp (a suhgroup of Po) which possesses the same strncture as the given abstract group G.
80
INTRODUCTION TO THE THEORY OF FINITE GROUPS
The permutation whICh corresponds to I is evid.ently the identiu£tl permutatIOn
(G,) I' = (G,) O,I = 0, Any other permutation of G' displl:tces each group element r-,mce It changeH U, into G,P, wInch dIffers from 0, unlesH l' =1. We Bhall now resolve pi into cycles. Let the carre· sponding ahsl,raet clemmt be of order T, thus PI =1
(852)
Bt'giuning with !tny element G of G, we know that it is clmngcrl by P' into (iP, which in tUl'lliS changed inbo GP2 ; the image of UP2 uncleI' 1" IS opa, and so on until we come to Upr-l wInch iH tmnsformcd mto Gpr, 1 e into 0, in YIrLue of (352). 'rhus we see that pi contains the cycle (0, Gp, Gpz, , ap'-I) . (353) of order r If H be an element of G not included m (353), we can i!:!olate a further cycle of P', namcly (Fl, HI', Hp2, ., Hpr-I), which is lIkewise of order T. This IJrocess is to be contmued unbil all elements of G have been accounted for We have then arrived l1t the resolution pi = (a, Gp, ..., (/PI-I)(H, HI', "Hpr-I) ... (L, LP, .., Lpr-I), wlIich shows that pi is a 1e(Jular permutatioll (p. 69). We may summarize our results as follows: TlTEORIUM
10 (Cayley),
W~th
each element P of an abstract
(fro'up
• G. Gt , 02' .. 0' Go, we can aS80c'iate a -re(Jular permutation
P'=(G t
(12
••.
Ou )
fltp U 2 P .. Ogp . 0
GROUPS OF PERMUTATIONS
81
'I'hp set of all tlWle pe1 rnutatwns forms a group G' (s~lbrlroup of Po) whwh ~s isomorphw with G; i.e. ~f P+-7P' anil Q+-+Q', then PQ+-7P'Q' When an abstract group G If! Isomol'plllC WIth a group G' whose elements are ('onorete ma1,hematical entitieH such as permutatiom; or matrices, we say that G' IH a faIthful representatIon of G in terms of permutatIOnfl or matrices, as the case may be. All proprrt16s of G fLre also }HlRSeHSeU by G'. Conversely, any information about G' which does not really depend on the specia,l naturc of ItCl c!rments, applies equally to G 'l'ho Idea of l'C'lwOhOlltmg au allClinwL group by oonol'oto elomentH iii in some l'cClpe<'ts analogo\lK to the UCIO of co-orclimttos in the trt'atmellt of gt'olllctrical problems; It carrieB with it the ClRmo analytic'al ac1vantageB and aesthetic blemishes 'rhe permutation group G' which, according to Cayley'i:l Theorem, mlLY be assocmtetl WIth an abhtract group G is oalled the regular representation of G. It is qUIte pOClsible that an abstract group may admit of more than one falthful repIeRentation in termi'l of permutatIOns which may even be of dIfferent degrees E g., consider the group
G . I, A, B, C, D, E
(3.540)
of Table 1 (p 12) To obtain its regular representatIOn we postmultiply the set by eaeh of the clements In tum, thereby transformmg it into the various columns of the multiplicatIOn table. In this way the regular permutatIOns can immediately be written down, namely lABODE)
l' = ( I ABC D E
= (l)(A)(B)(C)(D)(E)
A'=(~ ~ ~ ~ ~ ~)=(l A
B)(n DE)
B,~(I A. BCD E) '-""-(I B A)(O E D) -RIAl!JOD
82
INTRODUCTION TO THE THEORY OF FINITE GROUPS
, (IABCDE) C = C fi) D 1 B A = (1 C)(A E)(B D) D'
=(~ ~~ ~ ~ ~ ~) =
, (1IE ABC D E) D C B A 1
JjJ =
=
(1 D)(A C)(B E)
" (1 fj,)(A D)(B C)
On the other hand, as we have seen on p 69, the same group may hc faithfully represented by the SIX permutations of P3' whwh operateR only on three symbols 25. TransitIve Groups. In this and the following scction we Rhal1 considor permutations of a fixed degree n, i.e we shall be concerned ,nth subgroups G of the symmetrIC gronp Pn . The objects on which G operates wIll again be denoted by 1, 2,. ., nor by letters 0'.:, [3, .... DEFINl'fION 3. A group of permutatwns ~s said to be transitlve ~f it contams at leust one permutation wh~ch transforms anyone of the 11. letters ~nto any other letter. Othel wise the group is sa~d to be intransitive Note that thifl concept applies to groups of permutations only. Let a permutation which changes 0'.: Into f3 be denoted by PrxfJ irrespectIve of the effect It has on the remaining letters We note that PafJ-I changes [3 into tL.. There may of course be more than one Ruch permutatIOn for a gIven pair (1;, f3. Evidently the symmetric group Pn is transitive, as it contains all possible permutations, including the transposition ((/.13) which transforms (f. into f3. On the other hand, the group VI: (1), (1 2), (3 4), (1 2)(3 4) of order and degree 4 is intransitive because none of its permutations changes 1 into 3 Incidentally, thIS group is isomorphic with the group Va: (1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)
83
GROUPS OF PERMUTATIONS
which, on the contrary, is transitive Both groups are isomorphic with the four-group ((2.52), p. 49). The set of permutatiOlJS of G which leave the first symbol (1) unaltered forms a subgroup G1 , for the identical permutatIOn certainly belongs to thIs set, as doe~ the product of any two ot Its members. Generally, we denote by Get. the ~ubgroup of all permutations of G which leave the letter IX unaltered 11 A group oj pm mutations G of degree n ~s if, and only ~J, the subgroup GI ~s of index n rdative to G Proof. (1) Suppose that G is tran~itlve. By hypotheses it contains permutations THEOREM
tran,nt~ve
P W P I2,
",P In
which transform 1 into 1, 2, ..., prove that the n complexes GI P l1, Gl P I2 ,
It
••. ,
•
respectively. GlP ln
.
(3.55)
We shall •
(3.56)
are m fact a complete sy~tem of cosets of G relative to Gt , The complexes (3 56) are distlllct because the letter I is transformed differently by permutatiOns belonging to ilifferent eomplexes For let Q be any permutation of G whatsoever and suppose that Q changes I mto 0:, say. Since PIIX-I changes (J. back into 1, the permutation QPIIX-l leaves 1 mvariant and accordingly belongs to Gl> thus QPllX-lcGl>
QcGIP1Cf •
Hence the complexes (356) comprise the whole group, i.e Gl IS of index n. (ii) Conversely, if GI be of index n, let
G = GlRl +G I R 2 + ... + GIR n be the decomposition of G mto cosets relative to GI . It is easy to see that no two of the permutations B I , H 2,.
.,
Rn •
(3.57)
84
INTRODUCTION TO THE THEORY OF FINITE GROUPS
transform 1 mto t,he same symbol, for If it, an<1 R, were to change 1mto the same letter a, the permutatIOn R.R,-1 woulclleave 1 unaltered and would therefore belong to G1 By Chapter II, Lemma 2 (p 3-4-), tIllS would imply that G1R,=GIR" wInch is Impo~slble. Hence the permutations (3.57) may 1ll some order be tn,l>:en for the IlCrmutations (355) whose existence has thufl been estabhshed It'mally, If a and [3 be any two of the n lettelH of G, the pormutation P1rx -IPlfl transforms a into [3 becausc the iirst factor changes a into 1 and the second 1 mto [3. Hence G is tranHitive Rince i,he ()l'(Ier of a gronp iH dlVislblc by the mdex of any of itH flllhglollpfl (p 37), we have the COROLLAIW. 'l.'he OTdc1 of It flllnS1ti1,r (J1'01LP of pe1'm1Ltalions oj degree n is clt1Jisible by n. The conoept of tmnsitivrnesA may be generalized in the following way.
DEFINI'rION"
A (fro?!P
of pflrmulafions is smd to be
k-ply transitlVe if ~t contmns at least one pe1'm1ltation which changes any set of k (listinct 1>!Jmools «.1' «.2" ., CJ.], i.nto any other buch set [31> [32' ... [3,. (the a's need not be distinct
from the [3·s). A k-ply transitive group is a fortiori l-ply tranSItive, where 1< le. The symmetric group P" is obviously k-ply transitive, whcl'e k IS any of the integerll 1, 2, .. ,n. The number of distinct sets of symbols aI' a 2, •.. , ct./" i.s
v=n(n-1) ... (n-k+l), regard being hafl to the oreIeI' of the symbols III the set. Let H be the subgroup of all permutationR of G which leave the set 1, 2, 3, ..., Ie
unchangerl. By arguments which are analogous to those ullcd in the proof of Theorem 11 it ('1m be Rhown that the index of H relative to G 18 equal to v, there being one coset
GROUPS OF PERMUTATIONS
corresponding to each of the v Ret" of k letters. have the result·
85
Hence we
THEOREM 12. The order of a k-ply tran8~tive g10Up of deg1ee n 28 diviS2ble by n(n -1) . (n - k --1-1). Alternatively we might hUNe developed the concept of multiple tritIlt:lItlveness mduotively by using as definition what from our point of view IS a cnterion TUEOREM 13 s~mply tmns!twe
The g1'Mtp G 28 k-ply twn8itive 2f (i) G is and (ii) GI 2S (k - I)-ply transItive w!th 1B(Ja1d to the 6ymbols 2, 3, .. , n. E,g" the group G=A 4 011 p. 77 IS doubly transItive because the group GI , which C'onsists of the permutatIOns
I,
(2 3 4),
(2 4 3),
is evidently simply transItIve, as its leadmg symbol (2) is capable of being transformed mto each of the remaining symbols (3 and 4) 26. Prim.itlve Groups. Let G be a tran82t~ve group and suppose that it IS possible to arrange the n letters on which It operates III an array of l' rows and 8 columns, where . (3.58) 1's=n, 1'>1, s>l, thus aI' 2• ••• , hI> b2 , ••• , hs (1' rows), . (3.59) kI , k 2 , ••• , l~s
a
as}
in snch a way that the permutatIOns of G eIther permute the letters of anyone row amongst themselves or else interchange the letters of one row WIth those of another row (Ill some order) so that two letters which stand in chfferent rows of (3 59) are never transformed mto letters of the same row and vice versa. A transitive group whICh has tIllS property is saId to be im.prlm.ltlve, and the rows of (3.59) arc called irnprirnitive system.s. A group for
86
INTRODUCTION TO THE THEORY OF FINITE GROUPS
whICh no ImpnffiItive systems can be found is said to be primlt1ve. It should be noted that this distmction applIes to transItive permutatIOn groups only. Eaample 1. The group G ={(I 2 3 4)} which uunsIsts of the permutations
I,
(1 2 3 4),
(1 3)(2 4),
(1 4 3 2)
is imprimitive, having the irnprimitlVe system 1 31 2 4 ,
which, by the four permutations of G, is changed mto 42 131 241 311 2 4 ,31 , 4 2 , 1 3
1
respectIvely Ie.tample 2. It is quite pOSSIble that one group may posse:;s several setH of imprimitive systemB. Thus III the case of the four-group (1),
(1 2)(3 4),
(1 3)(2 4),
(1 4)(2 3)
each of the anay" 14 121 131 34 , 2 4 , 2 3
1
can serve as a s€'t of imprimitive systems. A doubly-tran8~t!Ve group is always prim~tive. the rows of the array
For If
(l), a2' •.•
bll bz, .• were imprimitive systems, the group could not contain a permutatIOn that transforms the l?au' all a 2 mto all b2 , which would he a contradiction to the definition of double transitiveness. In particular, all Rymmetrie groups Pn are primitive.
GROUPS OF PERMUTATIONS
87
27. General Remarks about Transformations. Let be an aggregate of ohJects x, y, . .. (Their nature need not be further speCIfied at present.) By an automorphism of r we mean an operatIOn A that associates with every x a unique image X,I wkuJk ~'s Itself an elem61d of r, in such flo way that different elementfl have llifIercnt images; we shall use the notatIOn A: x-+x A • • • (360)
r
If B bo another automorphism, the operatlOnAB is defined as the rosnlt of first performing A amI then B, thus . (3.61)
which lays clown the law oj composition for automorphisms. In general, AB =1= BA The Identical operatwn I mfl.kes every ohject correspond to Itself, thus I . x -------7- Xl ( = x). It plays the part of the wnt element in the Ret of all automorplusms of r. We shall now show that any sel of automorphlsms A, B,
C, obeys the a8soc~ative law applicatIOn of (361) we have
Indeed, by a repeatcd
XA(BO) = (X-I)BO = ( (X·l)B)O
and 1.e
for all :t, and therefore A (BG) = (AB)G.
This result allows us to prove by an indIrect argument that cOl'tam sets of elementl:! or operations satIsfy the group postulates, and in particular the associative law, Jt IS only reqmred to show that the elements in question may
88
INTRODUCTION TO THE THEORY OF FINITE GROUPS
be regarded as automorphIC transformations of a smtable aggregate r Example 1 If r be a fimte aggregate of n distinct ohjects, the set of antomorphisms conslstl:l of the n l permutations, whICh therefore form a group (p 64) Example 2 Let r be the aggregate of all points m [n]-space 'fhe lineal' transformatlOll
y, =a'1 x 1 +a'2 x 2 + .. +ClznX n of non-zoro dotermirutnt Ia", I may be regarded as an auLamorplmnn of r m win all the point x is transformed mto y. The Ret uf all thellc transformatlOns forms an (infimte) group. An important (mfinite) subgroup consists of all transformatlOlls which transform the sphere r 1 : X12 +.l'2 2 + .. +x1l 2 =1 into itself (orthogonal transfm'mattons) Example 3. The functions 1 z-l 1 z 1- z, z, l--=-Z' z where Z 1S a complex variable, may bc regarded as a set of automorphisms of the complex plane r (includmg the point co) into itself (see p 14). In all these examples the associative law has thus been imhrectly establiBhecl 28. Groups related to GeometrIcal Configurations.
Suppose we arc given a configuration of n points in threedimensional space, and let us consider rotations about a fixed point O. The set of all rotatlOlls which morely permutes the n pomts of the configumtlOn amongst theIIJ.selves, forms a hlTOUP whwh 1S isomorphic WIth a (proper or iml)roller) subgroup of Pn" The order of this group is indicative of the symmetry with whICh the configuratlOn 1S endowed. In the absence of any symmetry the group reduces to tho identical tranllformation.
89
GROUPS OF PERMUTATIONS
We shall now chscuss in more detail some caseR of special interebt. (1) DIhedral Groups. Consi(ler a plane lamina havlllg the shape of a regula,r polygon of n vertices, and suppose that the two sides of the lamina, arc completely ahke (FIgure 2 illustrates the case m whICh n =6.) There are 31--
.... 2
I)
5
.FIG
2
2n rotations, includmg the identical operation, wInch brmg
the lumina to coincIdence with itself. the lamina through one of the angles
Bor we may rotate
21T n about the hne through the centre 0 and perpenilicular to the plane of the lamma. These n operations may he denoted by I, A, A.2, . An-I, .., (n -1)--
00'
where A represents
!t
rotation through 2'lTln and
An=I.
0
(3.62)
A further operation consiflts 1Il reversing the two Rides of the lamma This may be accomplished by a rotation G through 71' about a hno jmning the centro to one of the verticeR It must be undorstood that, this line does not take part in any rotatIOn but is assumod to retain its (}
90
INTRODUCTION TO THE THEORY OF FINITE GROUPS
orientatlOn 1Il space rro fix the Ideas, let us take 01 as the axis of thc rotatwn O. We have · (3 63)
since 0 2 corresponds to a rotation through 27T, l.e. 0, and the 2n operatlOn8
OAAp.
(A=O, 1; 0=0,1, .. ,n-l)
. (3.64)
reprosent all possible automorphIC transformations of the lamina; for they allow any vertex to be brought into the Jlosition of any other with or without reversal of the two sides of the Jamml1. In order to complete the multiplIcation table we have to find a relation between A and C. A little geometrical consideration shows that
AC =CA-l, .
· (3.65)
whieh in VIrtue of (3.63) is equivalent to (.d 0)2 =1
· (3 66)
(The reader IS recommended to verify (365) by cl.:rawing diagrams analogous to those on p. 8.) We remark that the 2n elements of the group llilght equally well have been denoted by
AII-CA
(A=O, 1,0=0,1, ..., n-1),
. (3.67)
whieh on account of (3.65) iR evidently eqUIvalent to (3.64). Our results may be summarized as follows. The rotations which transform a regular n-gonal lamina into itself, form a gro1bp oj order 2n. 1 t M called the dihedral group oj' order 2n, and ~ts defining relations are An=02=(AO)2=I.
. (3.68)
It will bo recalled that the dihedral groups of orders 6 and 8 were introduced in (2.58) and (2.61) respectIvely, subjeot to the associative Jaw being confirmed. This has now been
91
GROUPS OF PERMUTATIONS
done lor all dIhedral groupH, Hmce they may be reprcsented as groups of rotatIOns The lhhedral group of ordcr {l is isomorphic with P3 (p (9)
It IS of interest to obtain allalytwal expreSSIOn:; for the opcratIOns of the lbheclral group of order 2n. 1£ x he a vanable rangmg ovel' the valueR 1, 2, .. , n, whIch denote the vertices of the lamma in counter-clockwi~e order, tho operation 11 is descrihed -t hy the congrllmwo ro1t\tion x j"'S:v + 1 (mod. n).
.
(a no)
Again, if x = 1 + Z, the image of .v undor th!' operation () it! gIven by x;O = 1- z. Thus we luwe x(l == 2 -.v (mocl n)
• (370)
All relations between the generatmg element 'I A awl may be derived from (3 (\9) and (370) . e.g., we have
(J
xAO""x(' + 1=2 - x + 1 =3 - x, x(Wj'=3_ (3- :1.')=.]';, (AC)2=1,
I.e. as in (3.66). (n) The Tetrahedral Group. Consider a regular tetrahedron WIth vertICes 1, 2, 3, 4 which is free to rotate about its centre O. There are twelve rotations wluch tmnsform the tetrahedron mto itself. Bar one of the vertlCes, say I, can be brought into the position of any of the four vertICes 1,2,3,4, and the sohd llan then be rotated through onc of the angles 0 or 27T/3 or 47T/3 about the line joining tIns vertex to the centre, wherehy the three farcH meeting at the vertex are cyclically mtel'changcd. Thus we have 4 x 3 operatIOns in all. Each operation of the tetrahedlal gronp pcrmutes the vet;,tices in some WI1Y, it is therefore Isomorpluc with a subgroup of P4 • Ii one vertex be fixed, the remaining .« The notatIon power of x.
.J.' I hu.~,
of ('ourse, nothmg to do With that for n.
92
INTRODUCTION TO THE THEORY OF FINITE GROUPS
three vertwes rI., fl, "1 cu,n only be permnted cyclIcally Henue the tetrahedral group includes all pos::nble eycles (a, fl, "1). We have seen on p 7H that those cycles generate the group A4 , which is also of order 12 'Thus the tetrahedral group cont;tins AI' but as It is of the same order as AI' It follows that the tetrahedral group ~8 isomorphzc lmth the altematin(! (flOUP Al (IiI) The Octahedral (Hexahedral) Group. The eClltres of th(' faceH of a regnlar octahedron may be regardocl aH tho vcrticcH of a eube (hexahedron), and convcrsply to c'vc'ry ('ubc we oan ins(,l'lbe an octahedron whoi:le Vel'tic('H he Itt the eentrcs of the bces of the cube. Hence Lhe two HolidH IHwo th(' same properties ot symmetry, 1 c if one IS traw,formcd into itself, so IS the ot,her. Thus the octahedral and he.mhedral (JI01tlJS nre identical, though only the first name is in common use In the prcgent dlSt:USSIOll of thIs group we find it more COIlvement to consider a cube than an octahedron Fir::!t of all, we see that the octahedral group consists of twenty-four operatIOIls, because each vertex of the cube may be brought into the posItion of any of the eIght vertwes, and when this Ims bcen done, the solid may be rotated through OIle uf the angles 0 01' 27T/3 or 4-rr/3 about the diameter through this vertex, givmg in all 8 x 3, i.e U, rotatIOns, lllcluding the IdentIty The foul' diagonals of the cube arc pcrmuted amongst themHelves when the cube is transformed mto itself. If a diagonal is carried over into itself, it either coincides WIth the aXIS of rotation. 01' else the operation interchanges its two end-points; in tIns case the aXIS of rotation is at rightangles to the d1agonall1nd the angle of rotatIOn 1S 7T We infer that no rotation of the cube can transform each 01 the four diagonals mto itself, for the axis of such a rotation woulrl have to be at right-angles to at least thrce dIagonals, which IS obviouRly impOSSIble. Honce two dIstinct rotations of the outahe(1J'al gronp correspond to two distinct permutations of the four diagonal::!. As thore exist only
GROUPS OF PERMUTATIONS
93
twenty-four permutatIOns of four objects, it follows that the octahedral gron1) ~8 ~8om01 phze With the symmelnc group P'l (IV) The Icosahedral (Dodecahedral) Group. Turn. ing now to the last two of the regular' polyhedra, we observe that the icosahedron and dodecahedron have the same propertres of symmetry For the centres of the twenty faces of an Icosahedron lU!Ly lie JOIned to form a regular dodecl1hedron, and converHoly, the twel,ro vertiecs of all icosn.heclron can be placeel aL the centres of the faccs of a suitable dodemdlCclron. ThlU, the iC(JoSrthrxlral (lncZ dodecahedral fl1'OUJl8 me irlenticrtl, and Pitlwr solid lIlay be lUwd t,o eXlLmillo the nature of the group clements. We rledde to t'iwosc t,he dodecahedron, with whic·h the l'('udcr of tIllS 140I'1OS of Ulllversity Texts is no douht more faulllmr, seeing rt, as he does, on the cover of each volume that ho tlLkes mto his ham\. First at all we remark that the doclecahmlml group contallls sixty drstinct operations. For any vertex may be brought into the position of one of the twenty vertrees, ami after the vertex has reaeheclits final pO'litlOn, the solid may be rotaterl about the diameter through rt These operatIOns canse eychc interchanges of the three faeel:! whICh meet at the extremities of the rhameters The possible angles of rotation are therefore 0, 27Tf3, 411'/3, whenee it follows that there are 20 x 3 (=GO) drstmet rotations (including the identrty) which bring the doelecahedron into ('omcrdence wrth itself. In Euclid's elal:!sical construction * a dodecahcrlron II:! derived from a cube in I:!twh a way that each of the twelve edgel:! of the cube is a diagonal III one of the faeeR of the dodecahedron. Conversely, If we start with a given diagonal, we can inscribe in the uodeeahedron one, aur1 only one, cube which has this diagonal as ono of its edgeR Since each face has five diagonals, it follows that five cubes can thus be rnscriberl. Any rotation of tho clodccahcdron into itself illduues a permniatlOll of the five CUbCl:I, Bearing
*
Elements, Book XIII, PropositIoIl 1'1.
94
INTRODUCTION TO THE THEORY OF FINITE GROUPS
mind that the edges of each cube arc in (1, I)-correspondence with the faces of the dodecahedron, the reader wIll have no difficulty m convinc1l1g hImself that no rotation of the latter (except the IdentIty) leaves all five cubes m their original position or merely tram;forms each cube mto Itself. Hence dIflercnt rutatlOns gIve 1'1,;e to dIiterent 'PerrnutationR of the five cubes, i c the dodecahedral group is isomurphic with a certain subgroup of order 60 of the ;;ymmetnc group Pfj. As we shall prove m Ohapter IV, Oorollary L, p. 123, that the alternatmg group Afj i;; the only subgroup of Pfj of or
1Il
Exaxnp1es
123456789)
(abOdef) oedfbrt into cycles. ]'inu the oruel's of the two given pcl'mutat,lOns (2) Express in terms of muLually exclusive cycloH (1) Ro,;olve (1) ( 469725813
Elml (Il)
(1) (abc ° •• k)(al) ; (Il) (a la2'" arxyblb. ° •• u,)(a,u r_ l (iii) (ala •... arxyzulb. .. b,)(a rCl r_l
al:L!JclC. . ct ) , e,). al,Lyze lc 2 • (3) Verify LhaL the pennut.atIOnH (1 2)(3 4) and (1 3)(2 4) commute. (4) Show that (ab. . lx)(xl1!~ ... A) '" (au " lrx j3 ••. AX), where a. b, ... , I, x, 11., {~, ••• , A are dlbtulCt symbols (5) Prove that a eyclo of degree In 18 even or odd aeeordmg as tn IS odd or even. (6) Show that a permutatIOn of d.ogroe n wlnch IS the pl'ouuct of r cycles (mcludmg those of ortier 1) is even or odd aceording as n - r is even or odd. (7) Prove that p. may be generated by the tranl:lpOSILlOllS (1 2), (2 3), .. 0' (n - 1, n). (8) Show that P"maybe gonorated by the cyole 0 (1 2 .n) and any tranSposItion T = (ab). (9) Prove that every rl'gular permutation can be expressed as the power of a cycle and. that, conversely, if 0 (1 2 • n), then 0' IS a regular pel'mutltt,iOll consIsting of d cycles of degree m' w1ll1ro d,., (m,s) £Inri m,'=ornld. =0
=0
GROUPS OF PERMUTATIONS
95
(10) Let Gt (~'-= 1, 2, .. " y) be the elements of an abstract group G and aSOlOClute WJth every element R of G the permutatIOn R which change!> 0, mLo R-IG, Prove that (J) the set of all R forms a groulJ G which is Isomorpluc WIth G, (n) overy permutatlOn 01 G ('ommutm, with every element of G' (of Cayley's TheOI'em, p 80), (1Il) a permutation that commutes With every permutatIOn of G belongs to G', amI vIce Vel'8ct. (11) Obtam tho group whICh transforms lj, rectangular lamina mto itself. (12) Prove {,hat, whcn the !I pormlltatlOwl of a tranfJitivo gIOup of degree n aro Wl'IttOll a~ precInct'3 of IllutlllllIy Elxclusivo eyrICH, they cOlltmn IHltw(Jell them (n - 1)17 lcLti\r~ HINl'S and ANSWERS J (i) (147 H)(:l6 5)(3 0), (u) (Ct (, elf) (b e), arrlors 12, '.1. 2 (i) (aflll , • , ki), (il) (IIryb l .•• b,aJc, •. , cd, (ill) (a,ycJc •.•. Ct)(xzuJb. "b,). 8. COIJsirlor Lho l'r!l'mutatJOllS G-'TO' whoro l' = 0, 1, .. , n - 1. 10, (IIJ) Lot G. ---;.-J(G,) bo sueh 11 perffiut[1tloll; If It COlumutcH With overy eIem€'nt, of G', WB mll&t haw f(G,P) =f(U,)P for Avery P of G. Put G, =[ 11 Tho four-group. 12, In the expansiOll of G relatJVtl to G! the letter 1 occurs III all pormutatlOns wluch do not belong to 6 ,• 1.0. (]- {fIn times; the same IS true for the othel' Jotters
CHAPTER
IV
INVARIANT SUBGROUPS 29, Classes of Conjugate Elements. In this and the following ChltTltcr we :;hall again be concerncn with general propcl'ticR of abstmct groups ])E~'INI'rlilN 1. '['wo I'lrnwn(s A (IneZ B of a group G are said to be conjugate unth res]Ject to G if then exists an element '1' in G s1lch that
'['-lAT -B
(4 I)
The element T, which need not be unique, is called the transformlllg element We note that the relation of conjugacy is (1) l'cjlexwe, i e A IS conjugate with Itself because I-1Al =A, (il) symmett'ical, 1 e if A IS conjugate with B, then B iR cOllJugate WIth A because (41) Implies that T 1-lBT1=A where Tl =T-1, (iiI) transitive, 1 e If A is
conjugate with Band B is conjugate with 0, then A is conJugate wIth 0 because if T-IAT =R and 8- 1B8 =0 it follows that (T8)-lA(T8) =0 We have already seen (p 22) that conjugate elementR are of the same order. The complex which conSIsts of all elements conjugate with A (mcluding A itself) 11> called the class of A and will be denoted by (A), thus (A) ""A +'['2-1AT2 +T 3 -1AT 3 +
It is of interest to determme the number of rostmct elements in (A), This is best done by enqlllrmg what clements of G oommute with A, if N 1 and N z be two such elements we have
AN1 =},\A, 96
AN 2 =N zA,
97
INVARIANT SUBGROUPS
and therefore
A (N 1N 2) = (AN 1)N 2 = (N 1A)N 2=N 1 (AN 2) = (N I N 2)A, i e the prodnct NIN 2 also commutes with A. Thus the elements of G whzeh commmte wIth a .fixed element A form n s~.bgroup N A of ol'de?' n.1> say; it is cnlled the normahzer of A. :For brevity let us put N = N land n =n I' Suppose that the expansion of G into cosets relative to N is
G=NT1 +NT 2 + ... +N'l'/l
('/\ =J),
(42)
whore h (=g(n) is the index of tho norrna!i7.or. A typiC'!11 element of N'I', may be written N'l'" whore 1'1 if:! (tHy element of N, Le. fllly clement which cnmmutpH wIth A. We have (N'P,)-lA(N'l.'.) =T,-I(N- I..:LN)'1', ='1" -IA'l'p irrespectivo of the element 1'1 chosen Thus all element>! which belong to the same coset in (4.2) transform A alIke. Conversely, two elements from chfferent cosetH tritlmfOl'lll A differently For if not, we shoulfl have a relation
T,-lAT, =T,-lAT" whence
A (T,TJ-I) = (T,T,-I)A,
•
which means that T,T, -1 commutes with A and therefore belongs to N, i e by Chapter II, Lemma 2, p 3-1" NT, =N'l',. Thus the index of the normalIzer of A is equal to the e~act, number of distmct conJugates of A. We collect these results in the following statement. THEOREM 1. Those elements of G which commute with a given element A, form a subgl'oup N of order n (the nOl'maZizer of A). If G=NT1 +N'l'2+' .• + NT,,, where g = nh, then the class (A) contlj,in\~ h cli,~tinct elements wkwh can be 'Umtten
T 1 -1A'l\, T 2-1A'l'2'
•. ,
'J',,-lAT h •
(4.3)
98
INTRODUCTION TO THE THEORY OF FINITE GROUPS
'VO mIght paraphrase thIS Important theorem as follows' let A be a fixe,l element of G and let X run through the g elements of G. Of the g elements X-lAX, which are thus formally obtained, only hare distmct, each occuning n times where n iH the ordcr and h the index of the normalizer of A (g=nh) An element A forms a dass by Itself (h = 1) if, and only if, its normalizer is Identical WIth the whole group (g =n), 1 e. A commutes with all elements of G Such an element iR caned an invarIant or self-conjugate element of G. In an Abelian group every clement Ifl invarmnt and the coneellt of dasHeH bccomeR Illusory. In every gruup, I is an invaril1nt element, Ie (1)=1,
because X-IIX =1 for every element X 'rhe varlOWl u1asses uf conjugate elements are 1rwtually c,l'chtsive, for if (A) and (B) hacl an clement in common, we should have an equation T-IArr =S-IBS,
whence X-lAX = (Sl'-IX)-lB(ST-IX) for every element X of G Thus every clement of (A) would belong to (B), and by a Slillilar argument we could show that (B) was completely contained 1ll (A) Heuce two dlstmct dasses have no element in common. Since eaeh clement of G belongR to some class, we have a decompositwn of the form (4 4)
where k is the number of dIstinct classes On equatmg the number of elements on both SIdes of (44) we obtam the important relation g=hl +h 2 + .. +h k ,
where that
a.
is the number of elements h. I g
111 (A L ).
(~= 1, 2, ..., k) .
(4.5) We repeat (4.6)
INVARIANT SUBGROUPS
99
and that h, = 1 if, and only It, A, is Rolf-conjugate. Tho following theorem IS a simplc applic.'1tion of these results 2 If a grO~lp is of order p"', where p is a the number of its self-cot/jugate e[('ments 2S a positVIJB multlple oj p. Proof. In (4.5) each term, hemg a factor of pm, if! either equal to nnity or clRc iR of tho form p~(fL";'- 0). If there be z terms of the formol' type, the /:,fJ'OllP haK z selfconjugate elements. Hence aftor suitable lWtJ'l'angellH'ut (+ 5) becomcs p'Yn=z+P/<+lJP.'-I- ... (O"j-L' /1/ ',. ). THEOREM
pr~me,
From tIns eguation It IS cvident that;:, nl11Ht bc a multiplo of p, and Hinec 1 is a belf-conJugate element and thereforo z ;<> 1, it follows that z is in fact a pol:ntIve multiple of p. 30. Invariant Subgroups. The cOllce}Jt of normalIzer is not confined to Ringle elements If K be a gIven complex 01 G, any complex of the form X-IKX IS said to be conjugate WIth K relative to G. Thoflfl clemcnt" of G which commute WIth K, 1 e. for wInch N-IKN =K or KN =NK, form a group N wInch is called the normahzer of Kin G. In analogy to Theorem 1 we have THEOREM 3. The number oj distinct COml)lexes conjugate wllh K is equal to the mdex of 2ts normahzer A complex which commutes WIth every element of G (N =G) IS said to be invariant or self-conJugate in G. In particular, If the complex be a subgroup we have DEFINITION 2. A subgroup H ~s called an invarIant (or normal or self-conjugate) 8ubgroup of G if ~t comrmltcs WIth every element of G, i e. if
HX =XH JOI'
et'el'Y element X oj G.
or
X-IHX =H
(47)
100
INTRODUCTION TO THE THEORY OF FINITE GROUPS
It should lJe noted that (47) IS a relation between complexes, i e. if 11 he an element of H, the conuitiOl1 of invarIance demanuH that X-IHX should be sume element H' of H, which, however, need not be the same as H. 1£ A ue any suhgroup of G whatsoever, the groups (flCC Chal}ter II, 'I'hcorem 2, p. 33)
X-lAX,
Y-IAY,
Z-lAZ, ...
arc culled tho conJugate subgroups of A Thus a 8ub(JrouLJ Z8 znmrumt zf, and only zf, zt coinczdes wzth all tts c01~i!lgflte 8ubgl0nps
'The two improprr ,mbgroups I and
Gam Hlvarll1nL bllbgroupH of G. In an Abelian groulJ aUsubgrOtlpS are wna1'iant We shall UHO Lhe liymlJoliealnota,tion
H-'
or
G:>H
to cxprcHH that H IS an lIlVa,l'lant snbgroup uf G 'Yc should like to rmphaslze once more that the can· ccpt of invariance is a relative one, 1 e. If H -.< G and He G1 c G c: Go, It mUHt not be taken for granted that H is an invariant Hubgroup of Gl,\ I on the other hand, we have of course H -< GI . If H be an inval'lant subgroup and
X=X l +X a +· . +X m any complex of G, we have X,H =HX" whence on summing ave I' i, XH=HX. (4.15) Using (47) and the rules (2.36) and (2.41) for the
H a.F. and L.a.M., we obtain the following results. (i) (ii)
if H1 -< G, if Kl , Ka1
H2~
G, .., then (Hl> H2l
••• )
• are znvariant complexes, then
is an invanant
subgrot~p.
~
G. {Kl> K2 ,
••}
'rIH' lllvariant Ruhgrul1}Js arrived at under (i) and (ii) might well be Improprr flubgl'OllpS. The followmg
INVARIANT SUBGROUPS TlIli:ORRlV[
4:
A ,9'ub(jl'oup of ~nrlcx 2
101 18
alway8 an ~n.
vanant subgroup Proof. Let /-I be a subgroup of G of index 2 so that there are only two cnsets of G relatIve to H. Thus if S be any clement of G that docs not belong to H, neither H8 nor 8H has an element III common with /-I. Hence G =/-1 +HS and
G =/-1 +8H.
It follows that H8=8H.
III ac1dItwn we have /-Ill "",II/-I for every element II of H (seo (2,]·1), II :32), tmrl honco alHo /-/(118) = (1I8)H. Smce ovory element .x of G IS rithor of tho forIU II or JIN, we Infer that /-/X =X/-I IH UllIvcl'Htdly true, Lc. /-/ Hl nn invariant BU bgl'onp Example I In the dihedral gronp
D2 " A"=02"" (AO)2=I, the cyclic group A= {A J of order n is an invariant subgroup of index 2
Example 2. The alternating group An i'l an invariant subgroup of P11 If A belong to an mvariant subgroup 1-1, so does every element of the form X-lAX, where X IS an arbItrary element of G, i.e. each element of the cla8s (A) IS oontained in /-I Tlms an invariant tiubgroup 'i8 composerl of comZJlete claS8es of G H=(I)-t-(Al+(B)+ . •
and conversely, ~f the smn oj several such classM sat~8jy the group postulates, it constitutes an Invariant subgroup of G Example 3. The twenty-fonr elements of P4 may be arranged according to chtl:ll:lOS tLS followl:l .
Ko=I
Kl = (1 2) + (I 3) + (1 4) + (2 :I) + (2 4) + (3 4) K2 =(1 23)+(1 2 .i)-f-(1 32)+(134)+(1·1 2} + (I 4 3) + (2 :1 4) + (2 4 3)
102 INTRODUCTION TO THE THEORY OF FINITE GROUPS
Ks = (1 2)(3 ,t) + (1 3)(2
It call be verified that the complex V=K o +Ks <=I+(l 2)(34)+(13)(2 4)+(L 4)(2 3) (·19) forms a group (see p. 1;2); tIns is therefore an invanant subgroup of P<j, 31. The Quotlent (Factor) Group. Let H be an invariant flllbgroup of G and consirler the deoompositIOn
G<=H81 +HS 2+ ... +H8 n
(81 =1),
• (,1.10)
whero (/ = hn Now consider the pl'Oduct (HS,)(HB j ). Smoe, by (47), 8,H = HS,> we find that (H8,) (H8,) = HH8,S) = HS,Sp
using (2 17), P 33 The last expresAlon is of the form HX and is therefore equal to one of the terms m (4.10), say
H(S,SJ) =HS]" Thus we 11ave . (411)
This result IS of fundamentalunportrl.llce as it allows us to regard the n cosets HS 1 , HS 2, .. ,HSn (81 =1) . (412) as clement::; of a group of order n Its multiplication table is typIfied by (4.11); the 1tmt element of this group is H because H(HS)=H2S<=HS and (HS)H = HH8 = HS by (+.7). Any complex HX IS equal to one of the cosets (4.12) even if X is not one of the elements 8. Thus the inverse of HS lllay be written 111 the form HE-I, because
(HB) (HB-l) =H2SH-l=H,
INVARIANT SUBGROUPS
103
and we lIuer that it, too, helongs to the set (4.12), though it may there have been expressed dIfferently. Fmally we remark that the associative law is fulfilled, having been generally establIshed for complexes in Chapter II ((27), p. 30) The group (412) is called the quotIent group of G relat~ve to H or the factor group oj H in G and is denoted by GIH. Its ardor IS (JIll (=n). It should be noted that equation (.1: JIl, which lead to the Idea of a quotient group, is essentially bas<'d on tho invariance of H The torlll quotient group 1\,11rl the symhol GIH will not be Ilserlllnleys H ll:! ftIl invariant suhgrouJl at G ExamZJle. In eXtLIllple 3 on p 101 the nosctH of Pt relative to the invariant subgroup V can be expressed Us follows' VI = V=I + (1 2)(3 4) + (1 3)(2 4) + (1 4)(2 3), V2 = Vel 2 3) = (1 2 3) + (1 :3 4) + (2 J 3) + (1 4 2), Va = Vel 3 2) = (1 3 2) + (2 3 4) + (1 2 4) + (I 4 3), V4 = Vel 2) = (1 2) + (3 4) + (1 3 2 4) + (1 4 2 3), V5 = Vel 3) = (l 3) + (1 2 3 4) + (2 4) + (1 4 3 2), Va = V(l 4) = (1 4) + (1 2 4 3) + (1 3 4 2) + (2 3).
The reader may verify that the Hix easets V, (~= 1, 2, . 0' 0) form a group, namely P4 /V, whIch is Hlonlorphic with P3 (see Table '1:, p 13), e g.,
V2 Va = V(l 2 3)(1 4) = Vel 2 3 4) = Vel 3) = V5 • 32. The Central. The aggregate of self.conJugate eloments of G, including 1, forms an Abelian group Z, which is called the central of G. For if A and B arc self-conjugate elements, we have X-LAX =A ami X-IBX =B for every element X of G, and therefore X-I(AB)X = (X-IAX)(X-IBX) =AB,
which proves the group property.
Also B-IAB =A, Le.
AB=BA.
Alternatively, the cp.nlwl mall be df'jined as the a(Jvreyate oj
104
INTRODUCTION TO THE THEORY OF FINITE GROUPS
those elements oj G whwh commute with all elements of G In Romc groups the central re(luces to the slllgie element I, while the oppmnLe extreme occurfl in Ahelian groups where Z = G 'fhe eentl'l11 is, so to speak, the Ahelw,n part of a group. 'lrT~e centlallS always an tnvarianl 81tbgroU1), because the equation X -lZX = Z is eel tainly satIsfied, smee 111 fact X-1ZX =Z for each element of Z As an applicatlOll of thel:lc Ideas we shall prove. THli:ORBlM r; A flronp oj mder pZ, where p is a l1rirne, is nlulillj.s Afiel/ltn 1'1'00f. By Theorem 2, the order of the central of !t group of order 11z is either pZ or 11. The former n,lternatIve is equivalent to sltying thltt the whole group G iH illentlCal WIth Z and is therefore Abchan. It reml,iufJ to conSIder the ease in which Z is of order p. '1'he quotIent group G/Z If! then of oHler p2/JI , l.e. jl, amI hence 1S a cychc group Thus we may write
G=Z+ZP+ . +ZPP-l.
Two typical elements of G may be written G1 =ZIP \
G2=Z2P "',
where ZI and Zz belong to Z and therefore commute with all elements of G Hence
0P2 =ZIZ2P~-i p. = Z2Z 1PH P. = GzG1> i.e. G IS Abelian. 33. The Commutator Group. Corresponding to any two elementsS and T of G, we define a commutator U by the equation . (413)
Evidently U =1 if, and only If, ST =TS. III an Abelian group all commutators are equal to the unit element
INVARIANT SUBGROUPS
105
Suppose that on making /-I anri'll rnn mdependently through the whole group wo obtain the commutatol'S U1> U 2 "
,
11",
. (.... 14)
It is possible that the product of two comnmtatOl'H C[~nnot itHelf he WrItten as a commutator, But III any caHe the set of all oomrnut(tlors gene/alps n ecrtam gl'oup
U={U 1, U2 ,
""
U",},
which ~8 wllcd the COInInutator group of G, If U=G, the gl'OUp G is calle([ a perfect group. TUEORIGII1 (j 'I'hc C01nrnutatm' (/I'O'lip i.s WI im'r:tnant sub· (/10UP, and ltS qlloticnt (//OUP L8 Abelwn; it i.s contained in eVe11/ znvm wnt 8ub(frollll 11'hich has an Abelian quotient (/1'0 lip
Pt'oof, (1) 1£ X he any elemeni, of G ftml If U be a typical commntator, \\00 have X-lUX =X-l S- l T- 181'X =Hl -l'/\-18l 'I\,
where 8 1 =X-W;'( and also a commutator, 1.0
'1\ =X-ITX,
Hence X-lUX is
X-lUX =U, which proves that U IS an invarh\nt subgroup of G. (il) The elements of GjU are the complexei:l UX, the commutator of two such complexes uan be Wl'Itten (UX-l)(UY-l)(UX){UY)=UX-lY-lXY, . (4.15)
in VII tue of the multiplication table for the quotIent gronp Now X-I Y-lXY IS itself an element of U and may there. fore be absorbed by iJw factor U on the right.hancl side of (415). Hence in GjU every commutator is equal to U, which IS the unit element of that group, i.e. GjU is Abelian. (iii) If R be any inVI\riant l':lUhgrollp of G whose quotiollt group is Abelifl,ll, then by the preceding argument
RX-l Y-lXY =R, H
106
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Ie R contains X- I y-lXI", which ml1Y be any element of U, thus R:J U. We conC'lUlle thlR flectlOll by provmg the foIlowmg result THEOREM 7. If A and B be two mvanant S1tbgl'OulJS whtch have only the unit element m common, then ever/) element of A commutes u'tth evel y element of B P/'Oof. COllfucler the comlllutatOl' U =.'1. -IB-lAB = (A -113- 1A)13 =A -l(B-IAB),
where A and 13 arc typical elements of A and B Hince A (Lncl Bare mvariant, A -1]]-IA I::l an clement of Band B-IAfl 18 an clement of A. Thull It appears that U il-J an element both of A and of B By hypothesIs it must then be equal to 1, Le. A and 13 commute. 34. Ho:mo:morphisllls and Iso:morphlsllls. The idea of Isomorphmm ha" ah'eady Leen mtroflueed on p. III If G amI G' are iliOIn01 plnc groups, we Ilh~tll wnte G~G'
The mapping of G outu G' can be uniquely l'everl:>ed, 1 e no two elements have the SlLille imlLgc But the typical fea,ture of the mappmg is the conservahon of stntct1t1'e, which IS most conCIsely expressed by (AB)' =A'B' . . (4.1G) (see p HI) Any mapping G ~ G' which satisfies conditIOn (4.16) IS eaUeel a hOlllolllorphlc mapping (or a hOlllolllorphislll) of G onto G' This mclnde':! cases in which two dIfferent elements of G may have the same Image in G' Thus m a homomorphIsm structure il-J retained but individuahty may be destroyell.* To mention a trivial case, we can map any group G onto the group whose only element is the number I , III fact, if A--+l,B--+l, 0--+1, ... arelatlOnofthe formAB=O
* Some
authors use the word
~somorphism
instoad of homo-
lUorphIll!ll and donote by SImple isomorphism what we call lAOmorph1snl. See footnote on pIll.
INVARIANT SUBGROUPS
107
carrIed over mto I x I = 1, WlllCh is eVIdently true. A rather less ohvious example Ii:! furnished by the alternating character (Chapter III, Theorem J, p. 7(1) of a group of permutationA, or by the homomorplllc cOI'reHpondence between a square matrix ~),nd itl; determinant 111 a group of matnres, where the multlllhcu,tion theorem of determinant:;, namely IS
I An I == I A I I B I, IH
the
re[\liz~),t[()n of
(4.1ti).
We shall noW" prove that if H i~ an inntri,lllL ,mhgl'oup of G, then G lH homomnrphin With G/H III fa('t, if we COl1l::trnct, the lIlu,pping
x--->- HX,
. (417)
where X is a tYJ)ical element of G, we see that C'orl'esponrling to a rolatIOn X Y = Z we lw,ve (HX)(HY) =H2XY =HXY =HZ, .
. (../. 18)
which shows that condition (4 ] 0) is fulfillerl It shoulll be obHerved that" 1Il accordance With our defirution (4.17), all clrments of G which belong to the same cosot relative to H have the same Image, for two sueh elements are of the form H1X and H 2 X rOf\pectively, where III and 11 2 are any elements of H. Henee by (4- 17)
HIX -->- HII1X =HX H 2X -+ HH 2 X =HX It is an interestmg fact that, in a sonse, aU homomorphisms are cqmvalent to those whIch arc generated by suitable invanant subgronps as in (4.17). fluppose we are given a homomol'phic lUltpping G -+ G' which m typified by X
-------';>-
X'.
. (4.19)
It is deal' that tho Image of the unit element 1 of G is tho unit clement 7' of G'; for ..in('o 12.: I, tho image of 1 must Ratisfy the equation (X /)2;= X', which has no solution o,})a1't
108 INTRODUCTION TO THE THEORY OF FINITE GROUPS
from ]'. Agam, HInCe XX-l =1 IS carned over Into X'(X-I)' =1', we mfer that (X-I)' = (X')-l, 1 e. X-I-+(X')-l.
. (420)
If the homomorphIsm (4.Hl) is not an isomorphism, l' is the Image of several clements of G Let
E=I+E 2 +E a + . -I-E! . (421) be the comple:c oj all elemPJ1fs of G whu:h me mapped on I' TV I' 8ltall show that E is an invm iant subgroup of G 11'01' if E, and lC) he ftny two cloUlcntfl of E, we have E, -+ l' and FJ3 -+ 1', whence hy (4.Hl) E,E j ----'7- ['I' =1', whIch shawl'! that the eomplex (4.21) IS a Cluhgrollp. Next, if E be a typIcfLl element ot E and X [lny r.lemcnt of G whatgocvor, we find, with the aid of (4 20), t,hat
X-lEX -------»- (X')-II'X' =1' Hence all elementfl of the form X-lEX belong to E, Ie. E is an l1lvarIant subgroup. Flllally, we shall prove that GIE IH IflOlnorphic WIth G' Consldel' the mappIng EX--)-X '
(4.22)
between these groups, wlJere X' is the image of X In accordance with the mappIng (4.19) of G onto G/. We have already seen that (422) leaves all structural relations unaltered. For on putting H =E in (4-18), we have (EX)(EY) =£(XY), which may be inteipreted as X'Y'=(XY)'. Moreover, distinct elements of GfE have distinct images 111 G'; indeed, if We hacl EX -------»- X' and EY ---+ X', our construotion of (4.22) would imply that X --+- X' and Y - + X' III the mapplllg (4.19), whence XY-I_--)-X'X'-l=l'; ie. XY-I would belong to E, and consequently EX = E Y. Thus the groups GIE and G' are isomorphIc, and the mapping (4.19) IS not essentially different from
X-+EX,
INVARIANT SUBGROUPS
109
whieh is of the type (4 17), being generated by the invarIant subgroup E We may ~ummanze these results as follows. THEOREM 8. IJ H IS an mvarzant subgroup oj G, then Gis hornornorphtc wIth GfH. In any hO'lnomarph!sm G---+- G', thol:!c clements oj G which are rnapprd on the umt element of G' form an tnvarinnt subyroulJ t. of G such that GfE 18 tsomorphic 'WIth G/.
35. Automorphisms. An llltm'oHtmg typo uf hlOmorphism occurs when tho group of ima,gPH ('olIwidc·s with the given group G DEFlNI'I'ION 3. A (I, I)-mapping {I} oj a IJI'OllJl G onto itselj whtch aS80ciatel:! With elJl'r!J element A oj G n llnllJZle tma(J1' A.p in G, (1). A-+A,p IS
called an automorphlsm
Ij
(.tiB)", = A ",B.p It follows from the generalre~ults of Chapter IIT, § 27 that the set of antomorphisms forms a group There are two types of automorphIsm: if X be a fixed element of G, the maI1ping
:=;:
A-+X-IAX(=A~),
whICh eVIdently satisfies the above eonchtlOn, is called an inner automorphism. In the caH8 of an Abelian group all mner autoIllorphisms reduce to the identical mapping (A~ =A). An automorphism which is not equivalent to the tranl:lformatlOn by a Rmgle clemont is Hairl to be an outer auLomorphism, E g, in the four-group ~'P=B2=J, AB""'BA, the mapping 1---+1, A---+- 15, B-+A, AB-+B:l( =AB) is an outer automorphism. Agam, in a nyolic group fA} of order 1n the ('orreHponclence
A-->-Ak:
110 INTRODUCTION TO THE THEORY OF FINITE GROUPS
constItutes an outer automorphlSm provIded k is prime to m. 36. Theorems on Quotient Groups. If H IS an lUvariant subgroup of G, the quotIent group G/H comllsts of all complexes (cosets) HX of which, however, only U/h are distinct (see (4.10), p. 102) If we wish to regard HX as a 8Lngle entity, i e. as an element of G/H, we shall tem· poranly employ the notation
(HX) 1Il
contmst to
HX =H1X +}J2X -I
+HhX,
.,
wInch stands for the collection of h element/:! of G The purpose of the Iollowmg throrrffis is 1..0 exauuue m more detail the homomorphism between G and G/H establIshed m Theorem 8 '1'he arguments which at first seem somewhat in volved. are not really dIfficult, If it is borne m mind that a relation between several complexes of the forill HX admits of two mterpretations, one involvmg elements of G/H and the other elements of G We pass from the former to the latter by omitting brackets. TUEOREM 9. If H-
A=HT1 +HT 2 + ... +HT r
(Tl =J),
. (423)
where r=a/h, the complexes (H1\) , (HT 2),
••• ,
(H'l\) .
. (42'1)
form a group, denoted by A/H, which from the very na.tme of its elements is seen to be a subgroup of G/H. (ii) Conversely, suppose that the complexes (4.24) arc
INVARIANT SUBGROUPS
III
known to form a subgroup of G/H so that by the UriterlOn on p. 33 we have • (4.25) Let us now dejine It complex A of G by equation (423). When the round brackets l\.re omitted in (4.25) we obtain
N=A, whioh proVOi':! that A i~ in fiLct It subgroup. The next theurOIn rleahl with the oa~e ltl whit'll Lhe suh. group A is an mVltl'HtuL subgroup of G. THEOREM
and (ii)
10
If H -< G and H -< A-< G, thcn (i) A/H -< G/H
~~~G/A
Convcl'8dy, every invanant 8ub(fl'oup
of G/H i8 of the fonn A/H, where A-< G. Proof. (i) As in the preceding theorem, we put (426)
A=2:.HT" A/H=L.(HT,},
where i runs from 1 to r. Since A is an invariant subgroup of G we have AX=XA for every element X of G, whence by the first equation (-1-.26) 'ZHT)( =2:.XHT,. . (4.27) On premultiplying this equation by H and remembering that (H2T,X) = (HT,)(HX), we Can tnrn it mto a relatlOn between cosetH, namely
L'Z(HT,)J(HX) = (HX)[2:.(H'I\)J.
.
•
(4.2~)
'rhis means that. A/H (second C'ql1lLtion (426)) commutefl with every element HX of G/H, Le. AjH is u.u invlLriant subgroup of G/H.
112 INTRODUCTION TO THE THEORY OF FINITE GROUPS
(il) 1'he correspondence
HX --+AX . (429) aSSocIates a dcfimte element of GIA with every element of GjH. For if HX =HX 1 we have XX1 - l eH and therefore also XXI-leA, wllCl1ce AX =AX1 • Moreover, (4.29) is a homomolplllc mappmg because (HX) (H Y) = HX Y --+ AX Y = (AX) (AY). Next, let us find those elements of GIH which are mapped on the 1mit clement of G(A, i.e. on A. If HX be such an clement we must IH1ve AX=A, ie X=A, where A is an element of A, conversely, If A e A, then
HA-+AA=A Thus those cosets of GjH which correspond to the unit element of GIA are precwely those WhICh form the group AjH By Theorem 8 tins gr'oup IS an lllvanant subgroup of GjH,ie AfH<, GjH, and the quotient group of the last two groups is isomorphic With the group of images 1Il the mappmg (429), thus
GjH AjH~GjA. (iii) Oonversely, let the cosets (HT,)
(~= 1,
2, ... , r; 1\ =1)
form an lllvanant subgroup of GjH, i.e suppose that [l:(HT,)l(HX) = (HX)['i.'.(HT,)].
By omitting the brackets and reversing the steps that lead from (4.27) to (4.28), we can write this equation in the form AX=XA,
INVARIANT SUBGROUPS
113
where A IS now defined by (4.26) Thus we sec that A is an mvarIant subgroup of G. TIlls uompletes the proof of the theorem Let us return to the general Mtdtiplicatwn Theorem (p 56) and apply it to two in/Jal Ulnt 8ubgl'o1l]J8 A and B of G On puttmg X equal to Band H er!llllJ to A in (..J:.8), we find that AB=BA,
whence we mfer from the Mult,Iplwatioll Theorem that llH' complex
C=AB is a gruup It IS, 1Il fact, an iuval'iant (proper or improper) subgroup of G becam;e X-lCX =X-lABX = (X-lAX)(X-IBX) =AB
=c.
The intersectwll of A amI. B WIll be delloted hy
O=(A, B); it is an invariant A and B.
~ubgroup uf
G (p. IOU), and hence abw of
TH:EORE:l.l ll. Let A and B be invariant subUlou]J1i of G. If C = AB and 0 = (A, B), then
C/A~B/O
and
C/B~A/D
Proof. Among the elements of C are those of the form AI (=A) and IB (=B), where A and B are any elements of A and B. These groups are therefore subgroups of C and, being invariant in G, are a fortiori mvariant in C (p 100). We saw on p 56 (eqnations (2.60) and (2 OS» i,hat if the decompositlOn of B relative to 0 he explcsRCd in the form (·.1,,30) B=DB1 +DB 2 + , +DB,,,
that of AB (= C) rolatIve to A IUay be written C =AB1 +AB 2 1 •• -I AB"
, (4.31)
114 INTRODUCTION TO THE THEORY OF FINITE GROUPS
In our present case the tcrm8 in (430) and (4.31) are the elements of B/D and CIA respectIvely. Thesc groups are therefore of the same ol'uer. We shall prove that they are, III fact, H:lOIDOrphIC by showing that the (I, I)-correspondence AB, ++ DB, (~=I, 2, '" n) represents a homomorphIC mapping For Jet (DB,)(DB,) = DB" be a typicall'elatIOn bctween the elements of B/D. On multiplying by A2 and remcmbermg that AD =A (Uorollary, tJ. :12), we obtain (AB,)(AB,) >==AB/" which is the eorrospollding relation in CIA Henoe B/D = C/A SimI. larly, it can 1e proved that A/D,....,C/B. 37. The Jordan-Holder Composition Theorem. It IS cO!Ulllon practICe III mathematics to study complex entities by rC'Jolvmg them into simpler components which III some sense are themselvcs irreduCIble Thus Integers arc decomJlosed llltO IJl'lmes, polynomials arc splIt up lllto ureduCIble factors rclatIve to a gIven field, and so on But such a rm:lOlutlOn is not of great value unless it p08sesses the .property of uniqueness III a well-defined manner. There are several methods of reducing groups. However, we must confine ourselves to one particular hne of approach which is due to O. Jordan and was subsequently elaborated by O. Holder. DEFINITION 4. A group which possesses no proper tnvariant subgroup t8 called a simple group. The (cyclic) glOups of prime order afford tnvial instances of I:limple groups, as they have no proper subgroups at all (Corollary 2, p. 38). Simple groups of composite order arc rather rare and conl.lllancl special interest (see Theorem 13, p. 121). DEFINl'fION 5. A 8ubgr01~p A is called a maximum invariant SUbgroup of G if there exists no tnvariant StGbgroup H other than G or A such that G!>- H >- A. . (4.32)
INVARIANT SUBGROUPS
115
By Theorem 10, (4.32) is equivo,lent to the statement that GjA has a proper invariant subgroup HjA. Hence we have the following result CRITERION. 1'he s7tbgrozlp A is a ma~:imum in'vanant sztbgroup of G tf, cmd only if, G/A i.~ a 8t1nple groUl). A group may well posReRs several maXllllum invarll1nt subgroups differmg hoth in Rtructurr. and order. It G/A be of prime order, t.lWll A is a maximum invariant sul/group. If G IS not fl, Rimple group, it luw a proper inval'iant Huh. group and therefore alHo a maximum inv11rianL I'JUbgroup A, Ray. Thus
. (4-.3:1)
G';>A'rI.
It may happen that A if! a Imnple group; if Hot, we have where Al IS a ml1X1mUm invariant Rubgroup of A. Contmumg m this Wl1Y, We arn ve at the l'eJ:mlt that 131'131 y group G whate11er possesse" a composition serles, i.e. a selies of subgroups (4.34) G 'r A'r AI';> .. 'r Ar _ 1 ';> Ar >- I, whele
G/A, A/AI>'"
Ar-ljA r , A,
(4- 35)
are simple groups, whtch we shall IXlll co:rnpositionquotient-groups. It should be clearly understood that wInle A. IS 11 (maximum) invariant subgroup of Ai-I' it Heed not bo invariant with respect to G, Al , A~, ..., AI - 2 If the orders of the groups (4.34) be g,
U,
al> .•. , a r -
1,
u"
. (4.30)
the orders of the quotient groups arc
a/Cl, a/aI> ..., (l'_l/Ur> ar •
. (4-.37)
reHpectivoly. These integers are known as the composi~ tlon indices. Notice that their procluot 1S equal to y.
116
INTRODUCTION TO THE THEORY OF FINITE GROUPS
The following fundamental theorem deals with the uniqneneRs property of eompositlOn series referred to at the begmnmg of tlus RectlOn THEUREM 12 (Jordan-Holder) In any two composition series ]01' a group G the composltwn-q1lOt2ent-groups me, apart from thetr sequence, isomorphw tn pairs Let us conRlder m more detaIl what this theorem implies. suppose th~tt
(I) and
G >- B>- B1 >- " >- Bs >-1 . are two OnIDIJOflltion serlOS lor G. the quotient groups and
GIA, AIA ll
(II)
It is then assorted that
,Ar_I/Ar> Ar
(III)
GIB, B/Bl' .. , BHIB s' Bs
(IV)
.
are Isomorphic in pall'S. In particular, It wonlJ follow that l' = 'I, and that the composItion mdices are, pOSSIbly after rearrangement, Identical WIth
alb, blb l ,
.. ,
b'_l!b" b•.
It was thIS last fact whICh was discovered by J ardan, while Holder later observed that the qnotient groups were not only of the same ordor but actually isomorphlC. Proof. Since the only group of order 2, namely the oyclic group (2' IS simple and posseBsos the 1mique OOlllposition series
we have a baRis for induction, and we shall henceforth aBSUIDe that the theorolll ha" already been proveJ for groups of order less than a Let us now return to the given composition BeneS (I) and (II). We have to distinguish two eases:
INVARIANT SUBGROUPS
(1) A=B.
and
117
In thll:l case the series (III) and (IV) become
G/A, A/All .., AT _ 1(AI' A, G(A, A/B ll
.
.,
(V)
BS _ 1(B s, Bs
(VI)
respectIvely. Sinoe we assume thn,t the theOlom holu/l for A, the quotient groups 011 the rIght of the vertICal line are isomorphio in pairs as they reprcllcnt composition seriell fOl' A. In particular we have 1'==S, and since Lhe leading terms of the two ser'ic!:! aro not only IHOlllOl'phio, hut actually iclentlCltl, we conclude that the C'ompletc I:lCl'lOS tLI'a equi. valent in the senile of the theorem (iI) kF B We have seen that an lllvariant subgroup of G which as an invariant subgroup (p. ll3), thus
IS
lJl
turn contains A
G»C>-A. But since A is a maximum invarIant subgroup of G, we must have either C=A or C=G. The former alternative must he rejeeted because B and consequently AB differs from A Hence we conclude that
G=C=AB. Puttmg D = (A, B) as m Theorem 11, we have
G/A.-B/D and G(B"-'AfD. . (J.3S) Smce by the Criterion on p 115 G(A anu G/B are simple groups, so are B/D and AID, i e. Dis a maximum invariant subgroup both of B and oj A. Let
D»D1 » . .. >--D t »1 he any composItion series £01' D whatsoever and consider the following composition series fol' G :
G>-A>-D>-D >-, .. >-D >-1 1
. (VII)
G >-8»D »D 1 » . ..
»D t »1
. (VIII)
1
and
118
INTRODUCTION TO THE THEORY OF FINITE GROUPS
havmg the quotient groups
G/A, A/D ,D/Dl>"" Dt
and
G/B, B/D ,D/Dv "" Dt respectIvely. The groups on the nght of the dlVislOn lme are identlCal in the two series, while those on the left are' iSOlnorphlc when arranged crosswise in pairs) as stated III (4.38). Let us wnte (VIJ)~ (VIII)
.
. (4.39)
to oxpl'e<Js that these two sarleS have the property demanded by the theorem. Since they have Lhe first two terms in common with (I) and (Il) respectiVely, we may appeal to the hypothesis of induction and state that (1)
e-'
(VII)
and
(11) '" (VIII)
Combining these results with (4 38), We deduce that (1)~ (IT). ThIS completes the proof of the IJordan-Holder theorem.
JJJxample I 'The altcl natillg group An .8 a maxtrnztm wvanant 8ubgioup of Pn. We have already seen (p 101) that p,. :>A n Since Pn/An IS of order 2, It follows from the Criterion on II 115 that An is a maXImum invarIant sub. group of Pn Let us finel a composition series 1Il the Cases where n IS equal to 3 or 4. (i) n=3 A com11ositlOll series for Pa can evidently be wl'ltten Pa :>A 3 ?I, . (4.40) the composition quotient group::! bemg the snuplc groups Pa/A a, As of pl'lme orders (composition indices) 2,3 .
(4.41) . (4.42)
respectively. (il) n =4.
The group V: I, (I 2)(3 4), (I 3)(2 4), (1 4)(2 3)
. (4.43)
INVARIANT SUBGROUPS
119
an lllvariant subgroup of P4 (example on p 102). As It consISts of even permutatIOns, it IS an invariant l:lUbgroup of A4 • Aga.m, overy element of V, other than 1, generates a subgroup oI Vwhich IS of index 2, !tnd is therefore invarifl,nt with respect to V. Hence P4 ;:"A j ;:" V;:" {(I 2)(3 4)} >-1 . (44.J.)
IS
iH a composition senes for P4 with composition indices
2, 3, 2, 2.
. (.t,45)
[CxamlJZe 2. The C'yelle group Cq whirh IS generaterl hy an clement A, where AG",,], POS!'lC'SflCS a eOllll'ositioll sC'r!('s oI the form . (..tAG)
The mIddle term IS a cyclic gronp of ortler' 3, n,nd the quotlOnt groups of (..lAG) are cyelic groups of orders 2 and 3 respectively. We observe that Lhe quotient groups of (4 M» Il,nd (4.46) are isomorphlo Thns we learn that a knowlcrlge of til£' COmp0E>ltlOn-quotient-groups docs not suffice for a reconstruction of the whole group. DEFINITION 6 A g101lll G is smd to be soluble if all its composition mdwes are pnrne. E.g, we see from (442) and (4.4.15) that the groups Pa
and P<1 are soluble. The decioion regarding soJn bility i~ often faCilitated IJy the fallawmg statement CRITERION. A [Jr01lp G is soluble if it contains an van ant s'ubgroup H such that Hand G/H are soluble Proof. If these conditiOns are fuUilletl we have
H>-H1;:" and
G/H>-G1/H>-
In-
;:..Hr;:"I •
(4.47)
';>-G,/H'(-H,
(4.4l:\)
where all eompoAitlOn indictlil in (4.47) amI (4..jJI} are primo. (It should be remembered that any subgroup of
120 INTRODUCTION TO THE THEORY OF FINITE GROUPS
GfH can be wntten 1Il the form AfH and that its unit element IS H) Smce by Theorem 10
~z-lfH_G G,fH
fG
'-1'
(Go=G),
we infer that
G>-G1 >- .. >-G,>-H>-H1 >- ... >-Hr>-I is a composition senes for G m which all Indices are pnme Hence G is 1Iolu11e.
38. GaloIs' Theorem on the Alternatmg Group. We shall prove in tIns sectIOIl that, provided n> 4, the alter. nating group An contains no proper invarIant subgroup Thu; is eqUIvalent to saYlllg that any invanant subgroup of An which does not merely consist of the Ulllt element IS equal to An We beglll by provmg the following lemma LE1IlItA 1. If an invanant subgr01tp H of An (n? 3) contains one cycle of degree 3, then H =A" Proof. There if! no loss of generality in denoting the cycle in question by (1 2 3). When n = 3, the alternating group IS generated by (1 23), and we have nothing further to prove. . Suppose now that n> 3 SInce H rs an invariant sub· group of An It contams every ]lermutatlOn of the form
8- 1 (1 2 3)S,
wheroS is any even permutation whatRoever. if
In particular,
S=(321e),
where k is an integer greater than 3, ,vo find that H contams the permutation (3 2 k)-I(l 2 3)(3 2 k)
= (1 k 2)
and consequently also its square, namely (12k),
(le=3, 4, ••.).
INVARIANT SUBGROUPS
121
By Ohapter III, Theorem 9 (p. 78) these special cycles generate the alternating group, I.e. H =A n We are now in a position to establIsh the celebrated result referred to in the headlllg of tIns section. THEOREM 13. When n > 4, An t8 a simple group Proof. tluppose that H is an invariant subgroup of An. (1) Let H include an clement of the form
II =ABO where A, E, 0, " A
=
. (4.49)
are mutually c:\clusivc eyC'les nlld
(n l a 2a:p I
...
am)
and Tn:> a.
The permutatlOll 8= (Il I a,P3)
commutes with all cycles of (4.40) except the first. S is even, H also contains the perml1tatlOns
Since
HI =S-lIfS = (S-lAS)BO . and HIli -1 = (S-lA8)A-l = (a 2a 3all 4 ., um)(ama m _ l " a4a3(t~nl) = (ala3am)(a2)(al) ... (am_I) = (ala sam)
(sec (319), p. 71).
Hence we infer from the Lemma that
H =A". From nOw on we can confine ourselves to cases in which the permutatlOns of H are products of cycles of degrees 2 or 3 only. (ii) Let H be a permutation of H involving at least two cycles of degree 3. There is no loss of genenl,lity in wriMng 11 = (1 2 3)(4 5 6)P, where P does not depend. on the first six letters. Choomng S = (2 3 4) as a transforming clement and noting that 8- 1]>8 =P, we [
121 INTRODUCTION TO THE THEORY OF FINITE GROUPS
deduce that H must also contain the permutatlOns HI =8- l HS '=" (1 3 4)(2 5 6)P
and
H 1H-l = (1 3 4)(2 5 6)(3 2 1)(6 5 4) = (1 2 4 3 6), contradicting our assumption that no cycles of degree greater than 3 should occur. (iii) Next consider the case in which H involves only one cyole of degree 3, say
H =(1 2 3)P, where P il:\ a product of mutually exclusive cycles of degree 2 so that P2=1. We cone'lude that H contains the per. mutation HZ = (1 2 3)2P2 = (1 3 2) and therefore, according to the Lemma, coinCIdes wIth An (iv) There rcmam8 the pOSSIbility that II involves no cycles of degree 3, but HI a product of transpmntioml Slloh a case does actually OCCllr when n = 4 andlead8 to the fourgroup Y which we discussed on p. 118 On the other hanel, when n> ·1, we can argue as follows: suppose that H
=
(1 2)(3 4)P
IS an element of H, where P is independent of the letters 1, 2, 3,4. If we let 8=(2 3 4) be a transforming element, we fiml that H contains the' permutations III =S-lHS = (1 3) (4 2)P and JI 2 =H1H-l=(1 3)(42)(1 2)(34)=(1 4)(2 3). Again, on taldng
1'=(145) we conclude that the permutations
INVARIANT SUBGROUPS
123
H 3 =T-1H2'1' = (4 5)(2 3)
and
HaHz-l,,",U 5)(2 3)(1 4)(2 3)=(45)(14)=(1-15) also belong to H Thull it appears that H contains a cycle of degree 3, whence it follows that H =An • 'l'hi!:! completC'1i the proof of the theorem COROLLARY 1. An is the only ,mbgrOll)l of ord!,l' ~nl contained ~n P1l' Proof. AllY flllhgroup H of that order if; IlC('CHSI1I'ily 1111 invariant subgroup of Pn ('Theorem 4, p. 1(1). H()jl('(\ the intenmction D = (An, H) is an invaril1nt subgroup of An. It follows from the l1bove theorem that either D = I or D=A" 11 n> I, H contums more than une even permutl1-
Hon uncl hE>nce has mOl'e than une eleml?nt in commun "it,ll
An (Theorem 8, p. 77). Thns An = (A", H), wheme All c H. But, as both groups are of the same order, we have in fact An=H. 2 Pn 18 not soluble when n > -1 simple when n > 4,
COROLLARY
An
HI
For since
Pn >-A" >-1 IS
a composition series for P,.,
2,
Its compositlOll indlCes are
in',
the second of which is certainly not prime when n > 4. The concept of soluble groups admits of a very remark· able application 1ll the theory of algebrau~al equatw1ts, where it il:! proved that the general equati.on of the nth degree can be solved in terms of radicals if, amI ouly if, the group P" is soluble. We have learned that thlH condition is fulfilled only when n=l, 2, 3, 4, which cxrlains why there are algebraical " formulae" for solving efjuatiouK whOi':IC (lcgrce does not exceed 4. On the other llf1ud, WI) are lead to tho conclusion thai no such formula!' Pl1n pOf\sihly oxiflt for tho quintic or equations of st,ill hi~hel' (1!lgl't'e.
124
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Examples (1) By usmg Theorom 1 (p 97) or otherwise, prove that the only permutatIOns of Pn which commute wIth a gIven cycle of dogree 11, are the powers of that cycle (2) Show that when 11, IS odd the cycles on 11, letters form two classe:; of conjugate elements lelative to Am each con· taming ;\(11, - 1)1 members, and that when n 18 even the cycles on (n - 1) letters are divided mto two clusses in An of in(n - 2)1 elemonts each. (3) Two classes (A) and (A-I), which are generated by mvorse clcmcnts, are called ~nve7'8e classes. Prove that (1) l1lverse clasflus contam the same number of elements and (11) It group of even ardor incluclcs at least one class which IS itlentwal With Its inverse class. (4) Resolvo Ai into classes of conjugate elements. (5) Show that if G=G 1 +G.+ I Go. the group {G 1 ", G .. , .•. , ao'} 1fl a (proper or improper) lUVarLant subgroup of G. (6) Let H be an invurlant subgroup of index 11, III G. If R be an element uf G such that Rt IS the lea'lt poslttVe power of R to he m H, prove that t 1S a factor both of 11, and of tho oruer of R. (7) Prove that the group A' "" B3 = (AB)' = I if! 1somorphic w1th p•• Hence or otherWise show that I +A'+B'A'B+ BADBD 1B an mvarmnt subgroup. (8) If the commutator a of A and B commutes WIth both A and E, prove that (AB)' =B'A'Oi,('+I). (9) If each element of 11 group commutes With evory conJugate, show that the commutator of any two elements commutes w1th both, and prove tlult the lllements whoso orders divide fl, given odd number e, form an InvarU1ut subgroup. (10) Prove that A. possesses 110 subgroup of order 6. (11) Prove that the commutator gl'OUp of Pn is A". (12) Find the contral of the quatormon gr01.tp (Table 12, p, 55), t\nd construcD tho quotIent group relatlVe to tho contral. Also obtain a compos1tion series fOl' tho quatermon group. (13) Do the same for tho clihedral gronp of order 8 (Table 11, p.55). HINTS and ANSWERS. 2. The l10rmahzers 1n An of such
INVARIANT SUBGROUPS
125
cyclcs are of orders nand n - 1 re~pectively. 4 1, (I 2 3) + (l 42)+(134)+(243), (132)+(124-)+(143)+(234), (I 2)(3 4) + (1 3)(2 4) f- (1 4)(2 3). 7 Let A -7- (1 2 3 4), B ----7 (1 3 2) 10 Such a subgroup would have to 110 IllvarIant, whICh IS mcompatible WIth the rcsult or Rxampl<J 4. 11. Consider the commutator of (1 2) and (3 4) 12. 1 + A Z , isomorphic with V; G, {A '}, {A z}, I. 13 The samo
CHAl'TElt V
SYLOW GROUPS AND PRIME POWER GROUPS If G be a group of order g, the order of any subgroup of G is a factor of g (Lagrangc'l:l Theorem, p 34) ThB questIOIl whether, con· versely, G possesl:les at leaRt one snbgroup whose order is equal to a preassigned factor of g presents great dIfficulties, which have not yet been surmounted. 'fo he sure, the case ot cychc groups is str!11ghtforward, and the answer is in the affirmatIve, as we flaw in Chapter n, Theorem 4, p, 38, But where non-commutative groups arc concerllerl, our knowledge is much lUore scanty We begin by proving l), lemma which is a particular case of a theorem of A Cauchy (see p. 129 below)
39. A Lemma on Abelian Groups.
LEMlIIA I If A be an J1behan g1'OUp of m'dm' 11 and ~f p be any pnme factor of a, then A contains at least one element of order p. Proof. The proposition is obviously true when a=p (a pr1ll1e numbet). We may therefore employ mathema;(,lCal munction and shall henceforth assumo thn.t a IS a composite humber divisible by p, By U!lapter II, Theorem 5, p, 40 A possesses proper subgroups. Let us Relect a proper' sttbgroup H of rnaXt'mttm order h, (h
126
SYLOW GROUPS AND PRIME POWER GROUPS
127
consider the product HT. Smee A i:1 Abelian, HT = TH whence, by tho MultiplICation Theorem (p. 56), it follows that HT IS a suhgroup of A It IS obvIOusly more comprehensive than H which, on the other hand, was supposed to be a maxImal proper subgroup. Hence we mUAt conclude that The product on the nght-hand side includes ht/d elomentA, where d IA the order of (H, T) Thus We obtain the rellttio!1
ad=ht. The left-hand slde lH divisible by J.). itfollowH that, 2i I t, Lt:'.
llut, Hillee (11, h)
~ 1,
'rhe element P ='1'8
is therefore of order 21, as required, 40. Sylow's Theore1us. Same remarkahle results concerning subgroups of a certain tYl)C were discovered by the Norwegian matllematician L. Sylow. i< DEFiNITION 1. Ij the order of a group G be divisible by pm but by no higher power oj p, where p is a zmme, then any slLbymup oj G of 01 de? lim Ul called a Sylow group correspondmg to p. THEOREM 1. Every group oj order g 11088e88e8 at least one Sylow {J?'otlP corresponding to each pnme Jar-tor oj g. Proo]. We shall again proceed by mathematical induc-
tIOn The truth of the theorem iii evident when g =2 (the snbgronp referred to 11eed not be a lwoper suhgroup) Let 9 = p"'g', where (V', p) = 1. H.esolvmg G int\) claSHes of conjugate elements (equation (4.4), p. OR), we luwc G = (AI) + (A 2) + ... + (A",), .. "Theorillnea sur les groupes rlp substitution" ", }J(atlt. Artn. v (1872), p. 581.
128 INTRODUCTION TO THE rHEORY OF FINITE GROUPS
and consequently (J=h 1+11,2+"
.+11,,,, (51) where 11" is the nurnbeJ' of elcments III the class (A,). It will be recalled that the normahzer of Ai is a group N, whose order is given by n.=g/h, (i=1,2, ... ,lc) (5.2) (Chapter IV, Theorem 1, p 97) We have to distinguish two cases: (i) Snppose thero exists an integer l between 1 amI lc snoh that h! > 1 and (lop p) = L It follows from (5.2) that n~ IS less than iJ and is dIvisible by pm (but by no higher power of pl. Hence, by hypothesis, the theorem is true for Nt ttnd we conclude that N! possesses a subgroup of order pm, which is of course also a subgroup 01 G and therefore a Sylow grollp corresponding to p. (n) Havmg d.lsposed of cuse (1), we now assume that for every vn,lue of the suffix ~ III (5 1) mthcl' (a) h, = 1,
or
(b) p I h,_
Terms of the type (a) correspond to self-conjugate elements, and we note that there IS at least one such term, since I is a self-conjugate element. Denoting the exact number of self-conJugate elements (1 e. the order of the central) by z (>0) ann collecting in (51) all terillS which are equal to 1, we obtain p"~g/ ~z +xp,
whence
pi z.
Thus we see that the ardor of the central is divisible by p. Since it is an Abelian group we learn from the lemm'a that G possesses at least one element P whICh commutes with aU elements ann. IS of order p, i e. X-lpX =P. The cyclic group P~{P}
is therefore an invariant subgroup of G and GjP is of order
SYLOW GROUPS AND PRIME POWER GROUPS
129
pm-lg'. Hence, by the hypothesis of induction, G/P contains a Sylow group of order 1)m-l. Such a subgroup can be written m tho form HIP where H is a subgroup of G (Chapter IV, Theorem 9, p. nO) uf order h, flA,y Thus we have
p m - l =hlp, i.e.
h=[)m Tho gl'OllP H arrived at ill thifl way IH thereforo a Rylmv group of G corresponding to p. As a corollary of tIns fundamentall'esult we HlmU deduro Cauchy's theorem referred to itt tho heginnmg of thiH ohapter, THEOREM 2 (Uauchy). If p be a pmnejactm oj the ordPr of a (It'oUP G, then G contains at least one element of order lJ Proof. Let H be a Sylow group of G of oHler pm. If H be an element of H other than 1, the order of H IS of the form pfJ., where fl->O. Thus If wo put
P=HPfJ.-r,
we have dIscovered an element of artier p, since
Pp=IIPI'=I, P=I=I. It is quite possible that G may have more than one Sylow group of order p"'. Indeed, if A be one Rubgronp of this order, so is X -lAX, where X is any element of G whatsoever (Chapter II, Theorem 2, p, 33), I e. all groups conjugate WIth A aro likewise Sylow groups. It might of course happen that some or all of these groups are identICal WIth A. On the other hanel, it will now be shown tha.t we need not look for Sylow groups elsewhere. THEOREM 3. All Sylow groups belongm{] to the .~alYte pnme are conj1l{Jrtte to one another. Proof. Let A and 8 be two subgroups of G of ordor pm.
130 INTRODUCTION TO THE THEORY OF FINITE GROUPS
Decomposing G relative to A and B (eqnation (273), p. 59), we have and ab ab g=~-rd;+'
ab '+d/
(5.3)
whero dp is the order of
Dp'=' (Pp-lAPp, B).
(5.4)
In the presont case a = b = pm and rJ =pmg', where (g', p) = 1. Hence, on dividing (5 3) throughout by pm, we ohtain I
rJ =
p'"
pll (4 -I- pm d + ... + -rr,:' . t
2
(5 5)
Since Dp is a subgroup of B, Its order, dp, must he of the form pi' (O
A more preCIse statement regELrding the nuw ber of Sylow groups is contained in the following: TII:EJOREM
It
If there be
e.'l:act~lJ
k Sylow groups of G
SYLOW GROUPS AND PRIME POWER GROUPS
131
correspond~nfj
to a pI ime p, then k ~:> an tntefjel' of the forln a factor oj the order of G Proof· Theorem 3 imphes that the number of distinct Sylow groups IS equal to the number k of distinct groups conjugate WIth one of them. Let A be a fixed Sylow group correspondmg to p and let N be the normalizer of A, Le. the set of all elements N of G such that 1 + px and
tS
N-IAN =A.
(tl.G)
If N be of order n, wo have (f=nk.
(5.7)
(Chapter IV, Theorem 3, p. 99), which shows that lc is a factor of fj. Since AIS a group, each clement A of Asatiidl(,s the relatiOll
A-IM=A (Chapter II, Lemma 1, p. 32). Hence A belongs to N , and, moreover, we infer from (5.6) that A is an mvariant subgroup of N, i e.
A-.< N.
(5.8)
Hence pm, which IS the order of A, is a factor of n so that we may wnte
n=p"'n',
(n',p)=l.
The last statement IS" j Ufltified because n, being a factor of g, callnot be divisible by a power of p greater than pm Let us now apply Frobenius' identity (Chapter II, Theorem 8, p GO) to the ~mbgroups A and N of orders 1)111 amI n respectlvely, thus
G=AP1 N+AP 2 N+ .. . + APrN
(G.9)
np'" npln npm if = (l~- + d + .. + _od;'
(;'.10)
and 2
where dp is the Ol'der l)f
Dp = (Pr-lAf'~, N).
• (5.11)
132
INTRODUCTION TO THE THEORY OF FINITE GROUPS
'There is no loss of generahty 3n assummg that PI =1 and therefore APIN =AIN =AN =N. using (58) and the corollary on p. 32. The cOl'responding term in (59) is npm
T=n. 1
No other term in the expansion (5 9) is equal to N. On dividing (5.10) throughout by n after substitutmg tor {j [rom (5.7), we get 1im
pm
pm
2
[ 3
I'
7.:=1 + d--+-l- + .. +~d .
. (5.12)
Now Dp IS a ~mhgroup of Pp-1API' which, like A, il'l of order lJrIl. Hence It follows that prJl/lZp is of the form pI'- where o<;:p,,-:;,rn. The essentIal step in the proof IH to show that, except in the first term, p., IS greater t}uw 0, so that
k=l+xp
. (513)
In order to arflve at this result, let us suppose that the term In (5.12) reduces to Ulllty, i e.
pm =dl\
~th
. (5.14)
By the same arguments as those which we llsed m the proof of Theorem 3 (p. 129), we come to the conclusion that D.\ =PI\-lAPI-., i.e , since DI.. is a Auhgroup of N, PA-IAPl\cN. .
. (5.15)
We have secm that pm iH the highest power of p which divides the order of N Thus N possesses one or more Bylow groups of order p'ffl We have in fact found two such sul)groups of N, namely A and PI\-lAP.\. On the other hand, A is invariant 1ll N and, by the Corollary on p. 130, is tho1'efore unique, LB. we must have PA-IAPI\=A.
SYLOW GROUPS AND PRIME POWER GROUPS
Remembering that N was defined as the we infer that
norlU~tltzel'
133
of A,
whence
However, thlS equation IS Impossible unless ,\ = 1. Hence all terms of (5.12) except the first are divisible by p, whwh at once leads to (5.13) This completei:! U10 proof of the theorem. As an application we shall prove the following: THEORl~M: 5. Any group G whosp, orclm' is oj the form pq, where p and q are pnmes s~lCh that p ¥= I (mod. q) and (II 1 (mod. p), is necessa1ily an Abelian group. Proof. The case in which p = q having already been dealt with (Chapter IV, Theorem 5, p 104), we shall assume that (P, q) = 1. Let P( = {P}) be a Sylow group of G corresponding to p. We know that the number of suoh subgroups is a divisor of pq and is of the form 1 +px. Since q;fi 1 (mod p), It IS necessarily equal to I, the other factors of pq, namely p, q and pq, being incompatible with the conditions. Thus by the Corollary on p 130 P is an invariant subgroup G of order p Similarly Q( = {Q}) is an invariant subgroup of ordel' q Hence we have
PQ=QP (see equation (4.8)), and by the Product Theorem (p. 50) the product of these two complexcs represents a group of oreIer pq/cl, where cl is the order of (P, Q). Since (p, '1) = I, the two grOUpR ean have only the unit element in common, i e. cl = 1 and therefore G=PQ. Again, by ChaptOl' IV, Theorem 7, p. lOB, every element of P commutes with every element of Q; in particular PQ=QP.
134 INTRODUCTION TO THE THEORY OF FINITE GROUPS
'Thus the pq clements of G can be written pCt.Qfl or QflPCt. (e<. =0, 1, .. " p - 1; f3 =0, 1, .., q-l), which evidently defines an Abelian group. 41. Prime Power Groups. This is the name given to groups of order lim, where p is a prime. All Sylow groups are of this type. Prime power groups possesl:! a number of interesting properties, one of wInch we encountered in Chapter IV, 'rheorC'ln 2, p, 99, where we saw that a group G of or<1e1' pm always contains at least one invariant, clement P of order p, We have then X-IPX=P, where X is any element of G, and the cyclic gr01£p
p={P} ~rhlS result may be generalized as follows' THEOREM 6. A group of order pm possesses at least one invanant subgroup of order p f', where 0 < f.L < m. Proof The thcorem is true when m = 2, SlUce the group is m that caso Abeha,n (Chapter IV, Theorem 5, p. 104), and any element of order p generates an invariant subgroup of order p. Thus We have a basis for induotion. Let G be a group of order pm (m> 2) Let P be an lllvariant subgroup of G of order p The qnotlCnt group G/P IS of order pm-l vVe may therefore assume that G/P has an invariant subgroup of order pIJ.- 1 • Such a group can be wrItten in the form
is an invm'iant subgroup of m'del' p.
A/P, where Ais an invariant subgroup of G, which must eVIdently be of oreler p". (Chapter IV, Theorem 10, p. III ) COnOLLARY, " All prime power grol£ps are soluble. For a gronp G of orner pm possesses an invarIant subgroup A of order pm-l, which in turn contains an invariant subgrollp or order pm-2, etc. Thus we can construct a
SYLOW GROUPS AND PRIME POWER GROUPS
composition series
G»A1»A z
135
» . »A"'_l»l,
in which all compositlOll mdices are equal to p. Example 1. There can be no simple group of order 200. For since 200 =5 2 x 8, the group contain.g k Sylow grOUIJEl of order 25, where Ie IS of the form. 1+ 5x (x =0, 1,2, ... ) lind a dIvIsor of 200. Since (le, 5) = I, we must have IJ I H, whICh is impossible unless ;:l, =0, i e. t,llCro is a unique invanant Sylow group of order 25. Thull the group 18 110t simple Example 2. There can be no 8Hnple group uf order 30. For if there were Auch a group, none of its Hylow group!! would be nnique. Hence It would have J +5( ~6) Sylow groups of order 5 comprising 6 x 4 ( = 24) elements of order 5. Also there would be 1 + 3 x 3( = lO) Sylow gnmp8 of order 3, and so the total number of element:> would exceed 30. ExaInples (1) Show that A4 has one Sylow gronp of orner 4 and four Sylow groups of order 3 (2) Prove that there is no SImple group of order 56. (3) If G be a group of order p2q, where p and q are primE'S such that q is leAS than p and nnt i1 factor of p~ - 1, provo thfLt G is Abelian (4) Prove that allY subgroup whose order IS I, power of p (a prime factor of g) 1S contained m at kast ono of the >':lyluw groups correspondmg to p. (5) Show that an invariant subgroup whosR oreler if! a power of p is contained in every Sylow gl'On}1 eorrosponding to p. (6) If overy Sylow gronp of G IS an mvmiant snbgrollp, show that G is tho direct pruduct of its Sylow groups. HINTS and ANSWERS. 2 Otherwise Lhoro woulrl 1m Olg-ht Sylow groups of o-rder 7 and soven of order 8. 3. PrOvo that both Sylow groups are lmiquo and USA Chaptar IV, Them'om 7, p. 106. 4, Use Frobemlls' expansion (Chaptl.'1' II, 'rhoorrm fl, p. (0) and apply it to tho subgroup m question and a Sylow W'otlp. 6. Uso Ohapter IV, Theorem 7, p. 106.
CHAPTER VI
ABELIAN GROUPS 42. IntroductIon. It IS eVIdent that the theol'Y of groups would be very much slmpleI if multIplication were com· mutative ElH well as associative In fact our knowledge of AIJehan groups is more complete and more systematic than that of any other class of groups, the most Important feature being that we possess methods which make a HUTveyof all abstract Abelian groups pOSSIble, a goal that IS fltlll quH'e remote m the case of non-commutatIve groups. Throughont this chapter It will be assumed that the multiphcation of all group elements IS commutatIve, 1 e.
AB=BA, whence by (1.7), p. 5
(AB)?n ""A?nBm.
(6.1)
Again, every suhgroup is an mvanant subgroup, and the product A1 A2 ••. Ar (6.2) of several subgroups Ai (~"" 1, 2, ... , r) IS itself a subgroup (Multiplication Theorem, p. 50). If no two of the groups A. have an element other than 1 m common, the product (6 2) may be wntten as the d~rect prOd1tct
Al x A2 X
•••
x Ar
(63)
(see p. 46). 43. Sylow Groups of an Abelian Group. Since all suhgroups of an Abelian group G are invanant, there is exactly one Sylow group corresponding to each prime that 136
137
ABELIAN GROUPS
divides the order (J of G (Corollary, p 130). (J
=P{1. P2r/, , .. Prr/r, 1
In fact, If
(p, =F p" If i =F j)
(6.'1)
then G has the l' Sylow group& of orders
AI> A2 ,
, Ar
PC/.I, pr/".
., par
(6.6)
respectively. Smce the nuruberA (6.5) are relatively prime, it is clear that no two Sylow groulJs have an element other than I in common. Henco the gronp
A,. is of order pt' , P2c/'" ., I p, C/., and Ul therefore identleal with G. Thus Al x A2 X
•x
G=A I xA 2 x .,xA,.,
(66)
and we have therefore come to the following conclusion: THEOREM
1
EveP"Y Abelian (JI01tp
t8
the rhrect product of
its Sylow (Jroup8 In the case of Abehan groups tho existence of Sylow groups may be estabhshetl from :first principles without ~he rather elaborate theory which was the subjeot of thl" precedmg chapter Let P be one of the prlIDe factors of the order (f of G, and let A be the corresponding Sylow group of order pC/.. ThIS group has the following characteristlC property: THEOREM 2, In an Abelian. fJroup the Sylow group oj order pC/. consists oj all elements of G whose order is a pOWM ofp Proof. Let R be the complex of those elements of G whose order is a power of P If X and Y be any two such elements, we have by (6 1) (XY)'=XfYf.
By taking for f a sufficiently high power of 71, we can make each faotor on the right-hand side equal to l, and hence It
138 INTRODUCTION TO THE THEORY OF FINITE GROUPS
also the product on the left. 'This proves that the complex R is a group. EVIdently, smce the order of any element of the Sylow group A IS a power of p,
AeR.
(6.7)
On the other hanel, we shall now prove that the opposite relation IS also true For suppose R were an element of R not contained III A The prodnct
A{R} would then be a group more comprehensIve than A and of an order which, lIke those of A ancl {R}, IS a power of p. This would be a contradlCtlOn to the fact that A is a Sylow group corresponding to p, I.e. one whose order is the greatest possible power of p. Hence we must have
RcA, and in view of (6 7)
R=A, which proves the theorem. 44 Basis of Abehan Groups. The SImplest AbelIan groups are the cyclic groups
Cm={A}, Am=I. We shall eventually arrive at the result that all Abelian groups can be wntten as direct products of eychc groups, and we would meanwhile remmd the reader of the few instances of this theorem which we encountered III Cha,pter II, § 16 Let us first of all consider a group which is in fact defined as a direct product of cyclic groups, thus
G = {AI} X{A 2} X... X{Ax,},
(6.8)
where A. is of order Tn,. ThIs is an Abelian group, and a typical element of G may be written
G "" A 1x,A l' ... Ax:"k
(0 ~ x, < Tn,).
(6.9)
ABELIAN GROUPS
139
Since the factors of a direct prorluct are independent (p 413), tIns expreSSIOn for G IS unique This is eqUlvalent La sayl!lg that a relation of the furm A{lA 2",
•
AI.e~
=1
(6.10)
necessarily implies that e,"-"O (mocl. tnt)
(z=l, 2, ... , k).
For a supposed equation
A 1"'IA 2"',
• • ,A,,~.
""(/ =..1 111'A 211'
, ••
Aly'
would at once lead La (6.10), where e, >=:tl- ,V" Ou ('om· paring the number of element8 011 both sldes of (U 8) we geb . (fLU) DEFINI'l'ION 1. If i,n an AlJe~!an group G there file. elements AI' A 2, •. , A k of orders lUI' m 2, .• ,ill" respectively Buch that every element G oj G can bp represented In th" j01 m
G-A x 1'" -
1
I.e:
2 ' .,
L
4."~ ( O<x,<m, ) h. i = I, 2,. "k
. (612)
Ht one, and only one, way, then AI' A 2, • ,A k me sald to jorm a baSIS oj G. Note that the umqueness of the representation Im1JIies that the basis elements are independent. Comparing (6.12) with (6.0) we see that an Abrlutn fJroup possesses a bastS Lj, and only if, it can be written 118 a dZ'lect product oj cyclic groups, a'l m (6.8) The set of basis elements (If it exist'll IS by nu meanl:l completely determined by the group. Thu; is indeed evident even in the case of a single oyelio factor, I!'Ol' if o be of order m, then
{O} = rOt}, where t iH any mteger prime to 1n, Again, let Tn =1'8, whero (I', 8) "" 1.
140
INTRODUCTION TO THE THEORY OF FINITE GROUPS
The group {OJ of order m ('ontams the subgroups {OS} and {or} of oruers l' and 8 respectively Smce (r, 8) = 1, these two groups can have 110 element ill common except I. Hence {osHor}, which may also be written as a direct product, comprises all 7S( =,m) elements of {O}, thus {O}={O'} x {or} . (6.13) Uotlvcrsely, consider a direut product
{A}x{B}, whose factors are of orders rand s respectively, where (r, 8) = 1. The element
C=AB
is of order I'S For we have Ors=A"B"' =1 On the other hanel, if m were an integer such that =AmBm =1, it would follow that Am = B-m. But 1 IS the only element wIDoh belongs both to {A.} and {B}. Honce Am=I =B-Tfl., i.e m must be a multlple of rand 8, and hence also of rs, which means that AB IS of order 1'8, thus
am
{AE} = {A} x {B}
. (0.14)
We can summarize (6.13) and (6.14) in the statement
C,s = Cr X C., if (1', s) .d . (6.15) It follows in particular that by a repeated applicatIOn of (6.15) we ean always arrange that the orders of the cyclic groups In (68) are powers of prtmes. Tn bet, if m=pocqfJrY • • we have em = C'PIX x CqfJ x CrY x . . . . . (6 16) 45. The FundaInental Theo:t'eIn.
T.u.rwr!EM 3.* Eve..y A.belian group is the dzrect product of cyclic, groups 'whose orders m'e pmile1'8 oj pl'imes.
'" a. A. Mtller suggests that this the(wcm be legUJ:ded as the fourth in order of importance, being preceded only by the theorell1s of Lagrange (p. 34). Sylow (p 127) and Cayley (p. 80), see G. A. Miller, H . .l!'. Bhchfeldt and L. E. DlCksOIl. Theory and Appl~cation oj Finite Groups (New York, ]916),
ABELIAN GROUPS
P1'oof.
141
(i) Let G be an AbelIan group of order g where
g=p{I.,p 2r:1.,
pfl.
We have seen that every Abelian group iR the diroct product of its Sylow groups (Theorem 1, }J. 137), thw;
G=A1 x A2 X
•X
AI,
where the factors on the l'ight.han(l side correspond to tho (liffel'enb prll1les dlvldin~ the OJ'der of G, It iH therefore suffiClent to prove the theorem for an Alwlmn g1'OUp WhOHO order, lIke that of a. Sylow group, iH it POWCl' of It prime. The orders of the cychc groups '>'illicit appear lU; hctOl'H m the du'ect product Will then antomatIcally he roworK ot primes (Ii) Let A be all Abelian group of order pIT', We wiHh to show that there cxmts a set of independent elements suoh thaL every elemonb X of A may be rcpresented as
X
=
A/I"A 2-'"
•••
A"xk,
The theorem lH evidently true when 7U = 1. For A IS thcn neceBsallly cychc We may therefore proceed by induction and assume that the theorem holds for all Abelian groups of order pIL, where iJ' IS less than m. Let Al be an element of maXImum order, If Al HI of order pUl, then A={A 1 } and there is nothmg further to prove, Suppose then that the order of Al is pm" where Jill
f-="m 1 <m, We note that X pln , =1 for overy clement of A. (Iii) Consid~r the cyclic subgroup
A1 ={A 1}· The
quotien~
gronp A/AI is of order
pllHlL1•
It consists of
142 INTRODUCTION TO THE THEORY OF FINITE GROUPS
all complexes AIX and by hypotheslB may be derived from a set of baslfI elements
AI B2 ,A1B g, .. ,AlB/"
· (6.18)
whose orders WIll be donoted by · (H.19)
respectively, where (6 ?O)
Since Al is Lhe ullit clement of A/A], this means that
(AIB,)pml=AIR/m'=Al
(~=2,
3, ... , le),
i c pm! is tho exponent of Lhe least pOAiUve power of B. that lies in AI' Hence we may pnt · (6.21) It follows that J)m, cannot exceed the order pit of B. and, in VIew of (617), is certainly not greater than pm,. Hence we have m. <,j, >-;m I (i = 2, 3, ... , k), and on USlllg (6.20),
· (6.22) Raising (6.21) to the (pm,-mt)th power, we get B.prll, =1 = AIpmdtlipm,).
SiDce AIlS of order pin" It follows thai t,/pm, is an mteger u., so that we may write Thus if we put
A,=A1-u,B.
(~=2,
3, ... , k),
(6.23)
(i;., 2),
(624)
we have
Alm£=AI-tiB/mt=I by (6.21)
Since AlA, = (AIAI-tll)B. =AJB.,
ABELIAN GROUPS
143
the basIs ((US) may he written in the alternative form
Al A 2 , Al A 3 ,
AlAI" . . (6.2,15) and we note that, according to (6.24), A, IS of order pm•. (iv) Stal'tmg from tho basis (6 25) of A/All we shall ')
deduce that the elements • ((),26)
fOI'Ill a hasis of A. 111(leer1, if X he fLlly elemont of A, the oomplex AX is an clement of A/AI alld (mll j,herofo/'() lin exprcsl:led Ih tOllllR of (6.25), thus
i c.
AlX = (Al A 2)J,(Al A3 )x, ...
(AlAIY'~,
It follows that
X (Al'A.)"'L .. 11 ,r"'.)-l IS an element of AI' SILY AI"'" Hence X
= Al"'·A 21", ••• AI,"",
It remams to show that the eleruentf! (6 26) are mdependent. Suppose we had a reJ[l.tlOn
A{.A{, .. AI,ek =1.
. (027)
Premultiplying by Al we should fiucl that
(A l A 2)e, (A1A 3)C,. • (AlA~)ek =A I Since the complexes (6.2fl) are by hypothesis independent" we infer that Ci (i;;;. 2) is a multiple of the ol'uer of AlA. (when regardet] as an element of AfA l ), thus
e.=z,llm j
(i;;;.2)
. (621\)
Substitutmg in (6 2i) and ufling (6 24), we oMain A{l '"-I,
whence finally e1 =z!l)"".
Hence an cqna..tion of the type (6.27) il,\ im))O'\slble l1nleRS
144
INTRODUCTION TO THE THEORY OF FINITE GROUPS
each factor IS separately equal to the umt element. completes the proof of thc Lheorem.
This
COROLLARY. In every Abellan group the order of any element is a factor of the maximal order; i e there exists rtn element of Older h such that every element of the group ~s a solutIOn of X" =1. Proof. The proposition has already heen established for AbelIan gronps whose order is to power of a prime (see (v 17»). In the general case let G be written as the direct product of its Sylow group:;, thUH
G=Al xA 2 x ... xA I •
•
(62fl)
If Iff iI. he an element of ma},.imal order fLl-. 1ll AI-., we know that every clement X A of Al satIsfies the equatIOn XI'I-. ~ I. Hence it follows that 1ff = M I M 2 •••
is an clement of order
fL,
~Ml
where
fL = fLI/'/"2
•
fLl'
On the other hand, every element X=X I X 2 •
Xl
of G satisfies the equation X'" =1. 46. The Elementary DIvisors and Invariants of an Abelian Group. It appears that there IS consIderable latitude in choosing the basis elements of a given Abolian group. But while the basis elements themselves are not uniquely determined, it WIll be shown that their orders possess invarIant propertIes which completely specIfy the structure of the group. We have already seen that the orders of the basis elements are powers of primes If the basis is constructed in accordance with Theorem 3. THEQ!lEll[ 4. ~f the orders of the busts elements are powers 0/ primeS, they are called the elementary dIVIsors
145
ABELIAN GROUPS
of the Abehan (Jroup. They are completely dete1'rnined by the g10Up itself and, alJalt fmln thel?' sequence, do not depenrl on the particular basis chosen. Proof. (i) It is sufficient to l)]'ove the theorem for a glOllp whose order is a power of a Imme, Rlllce the elementary dIVIsors of any Abelian group arc those of its Sylow groups taken together. (ti) Let G be an AbelIan group of order lim and suppose that we have two sets of baSIS elementfl . «(l.30)
of orders respectively and . (G.31)
of orders
In order to simpliiy the notation let us put, fLi=pm l (~=1, 2, ..., k); V,=pflJ (J=J, 2, ..., t). (6.32) The baflis elements ma,y be arranged m such a way that
respeotively.
fL1
> fL2 > ... );. fL~(> 1)
v1
>V 2
and
>
.
(6.33)
.;;;.vt(>l).
(6.3.!)
We have of course pm =fLJJl'2' •. fLI, =V1V 2 • • V t (see (6.11), p 139) It is required to show that k=t and fL. =1/. (i = I, 2.•.., k). Suppose the theorem were fnJse and let fLt be the first number m the set (6.33) to ditler from the corresponcImg number m (6.34) ; thus we have VI =fLlJ V 2 =/J,2, • , ., Vf _ 1 =fLf-I'
(G.35)
but VI*' }if' To fix the ideas let vf>fLf'
(If f = 1, equations (8.35) are to bc omitted.)
• (11.36)
146
INTRODUCTION TO THE THEORY OF FINITE GROUPS
(hi) In every Abelian group G the totality of elements which can be represented as the /h,th power of some other elem.ent, fortUs a subgroup H. say, for d
X = UN,
Y = VI'-/,
we I1n,ve XY=(UVtf •
The definition of H is independent of any basis of G. On the other hanel, a basis of H may be constructell simply by raiRing all hasis elements of G to the !-Lith power Any element ",hose order IS a factor of iLi lH thereby reduced Lo the unit clement and should be omItted. Genemlly, an elem.ent of order /h, is changed mto one of order /h,J!-L/. Since by (6 32) we have fJ;, I !-Li if, and only if, /h,
form a basis of H
• ,Ai_II'f
They arc independent and of orders !-LI!f1.t' !-L2!0t,
." /ht-lJ!-Lt
respectively, whence the order of H IS found to be given by l~ = (01!-L2 •.. f1'f_l)!P,/-l.
. (6.37)
It 18 equally true that the 0/th powers of the elements (6.31) would also provIde a baSIS of H provided we retam only those amongst them whose orders were originally greater than /hf. Although we oannot tell exaotly whtOh clements of (6.31) are thus involved I It is clear from (6 34) and (6 36) that B l 1
arc certainly to be included elements are of orders
111
snoh a basis of H.
vl/p,t, V2J/hil .• ,vtJj.Lt
respeotively. Thus it would follow that h;;. (v1]'2 .. Vf-lYt)J/h/.
These
ABELIAN GROUPS
147
Substitutmg for It from (6.37), we obtain (JLlJL2 ., JLI_l)/JL/- l > (v lv 2 . . vl)/p/, whence by (6 35) 1 > vl/JLI> in oontracliotioll to (636) ThuH we must oonclude that the sets (6.33) and (6.31) are, in fact, identioal DEFINI'J'JON 2. An Abelian group of order pm i8 saM to be oj type (ro l , III 2, • In/,,) if it iii the dir('ct product oj cyclic (}1'OU]J8 oj orders 'J
(0.37)
where i.e.
'in l
>m 2 >
if
(6.37) are the elemental y d~Z'i801'8 of tlu;
... ;;;.m/c>O, rn l +m 2 +
+mk=In,
(6.38)
'J1OIlIJ.
Thus there are I~A many different AbelHLIl groups of order pm as there are pattitions of m satiAfymg (638). E.g, there ltl'e three Abelian groups of order 8 (=2 3) corre· spomling to the types (3), (2, I), (I, 1, 1) (see Chapter II, p. 51) respectively Suppose next that the ordel' (J vf an Abelian group G involves more than one pnme, say g =pa'1br~, •.. so that G has Sylow groups of orders pa, qb, r C, • • • respect· ively. The elementary diVisors of G may be arranged in an array * thus pal P"',
q>~::,
.. (a 1 ;;. a2 ;;;. ..,
.
1',1·, .•
.....
(b l
>b 2
(C 1 >C 2
...
at +a 2+... =a)}
> .. ,b1~b2+ .. :b) . (6.39)
>.
,c1 I-C\l+"'-c)
. ...
Eaoh row oonta,ins the element£l,ry divisOlS of a Sylow group, and G is the direct product of all the cyclic groups >I< Such ali array is not, m general, rectangular. Bllt, If tl(,Slrf'U, empty space~ may he filled up with umts in order to make Lhu row~ equally long.
148
INTRODUCTION TO THE THEORY OF FINITE GROUPS
whose Oluer~ are hsted 1U (6 3\)) It is impossible to carry the decompositIOn of G into cyclic groups any further, and in that respect the elementary divIsors of an Abelian group correspond to its ultImate constituents. On the other hn,nd, if we waIve the condItion that the orders of the cyclic factors should be powers of primes, several of the cyclic groups whose orders are prime to one another mn.y be COrll bined into one. Thus by applying (6.16) we can gather the cyclic groups whose orders are given in the firflt column of (6 39) into a smgle cychc group C Il ,=C p a, XCqb,
xC,a, x
Snmlal'1y, corresponding to the second column of (639), we get
Cn,=C 1I", etc. Since generally
al
X Cql), X
? (~2' bI ;;;, b21
C,.a, x
°1 ;;;' £:2'
' ••
we have n 2
I nl> and
n, I n'-I' 3 'l'he numbers n l , n21 obtatned by 'multtply-ing the numbers tn eacA column of the array of elementary divisors ate called the invariants" of the group G, 2'hey (lre oha1aoterized by the follow.ng propertMs DEFINITION
(i) G = Cn, X C", X .• , (g =n1n 2 •• ) (ii) n, I nH (t ~2, 3, ...). The invariants are 1~niq1~ely determined by the elementary divisors and, conversely, If we resolve each mvariant into prIme factors, we recover the array of elementary divisors, Thus the invariants &1'e completely specified by the group itself and do not depend on the choice of basis. Vvhen the order of the group is the power of a prIme, the elementary divisors are the same as the invariants. But when the order is divisible by at least two prImes, the number of Invariants III less than that of the elementary divisors.
* The nomenclature IS not completely standardIzed, some authors
the term 'invauanLs 1 for what we have called elementary divisors, or mchscl'uninately for both concepts.
USIng
ABELIAN GROUPS
149
Example 1. An Abeltan gr01~p oj order pm and type (mll m 2 , • mJ.-) contams pi\. - 1 elements of order p For let AI' A 2 , AJ.- be a basIs of the group and cOllflldor the equation Xv =I. If 0
.,
0
0
.,
X =A I "lA 2""•
.•
Ak"'l.
.
(6.40)
be a solution, we have
1 =A I Vlll,A 2Px,
•..
A k 71Xk ,
whence on account of the independence of the basis elements I.e. x,=O (mod. plnj-I)
(i=l, 2, .. ,le).
For a fixed value of i there are 1J solutions compatible with the conditions o<..S:t:. 'p!1l.-I, where ,\ =0, 1, 2, p -1. Substituting in (0 '10), we sec that the equatIOn Xv =1 is satisfied by pk distinct elements, including the unit element. Thus the gl'OUp contallls exactly pic - 1 elements of order p Example 2 Find the Sylow groups, elementary dzvisors and invanants of the yroup .A 30=B12=1, AB=BA. 0
•
0'
The order of thIS group is 360( =2 3 x 3~ x 5). Its Sylow groups are of orders 8, 9, 5 respectIvely, and may be speCIfied as follows : {B3, A15} of order 8 and elementary divisors (4, 2), {AIO, B4} of order!) ann. elementary divisors (3, 3), {A6} of order 5 and elementary diViSor (5). The invariants are accordingly
n l =4x3x5=60, n 2 =2x3=6. Examples (1) Prove that if the order of an A.belian group is not clivisible by a square, It must be cyclIC.
150
INTRODUCTION TO THE THEORY OF FINITE GROUPS
(2) Show that the greatost 111variant (n 1 ) may be charactel'lzed as tho maXImum order, or alternatIvely as the L.C.l\1. of all orders 111 the group. (3) Show that the remainder classes prImo to 24 form an AbelIan group of order 8 (= ep(24» and Lype (1, 1, 1). (4) :B'md tho Sylow groups, elementary dIVIsors and mvarIants of the fullowmg Abehan groups' (I) AIG=B~=l, (11) A20=.Bo=O"=J, (ill) AH=I, AO=B15. (5) By comparmg tho number of goneratmg elements on bobh I:lIdos of (6 16), deduce the well-known formula
ep(rn) =m(I-~)
(1 - ~) ....
(6) I:lhow that an Abellan group of order g has at loabt one subgroup whu
BIBLIOGRAPHY BURNSIDE, WlVL, Theory of (J"OUpS oj jlmte 01'([1'1', 21111 rd" 1911. CARMICHAEL, R, D., Inll'OdUl'tlOn to the theory (1 {/rOll]J8 offlillte
order, lll37 H., An Introduction to lite tM01Y of (JrUllJi.9 oj jini/I' ordm', lU08. MATllI~WKON, L C, Etementfl/!/ theory of g1'OUp9, 1U:IO. MILI'lTIR, G. A., BI,IIJIU'ffiLD, H. Ji", DWKSON, L l
INDEX The numbers lefer to the pages Aitken, 9, 70 AIternatmg churacter, 74 Assomate complexes, 35 Abbociatlve law, 2, 3, 9, 12, 14, 87 Automorphism of a group, 109 Automorphism of a set, 87 AXIOms, 2, 9 Bilsis, 139 Cancelling rule, 9, 12, 18 Cauchy, 73, l2Q Cayley, 11, 7fl, 80, 140 Contral, 103 Circular pormutatIOn, 66 Class of elements, 96 Classes of permutatIOns, 70 if Classes of resldnes, 15 Closure, 2, 9 Commutative elements, 3 Commut,ator, 10.! Commutator group, J05 Complete set of reSidues, 15
Complex,2H CompOSitIOn mdicoeH, 113 CompositlOn-quotiont-groups, 115 CompOQltIOu serios, 115 Congruent 1Utegel~, 15 Conjugate elements, 16 Co~et, 35, 37 Cyclcs,6fi Definmg relatIOn, 44 Degree of p"'fmutatlOn, 62 DIroct produot, 'J 5 Elementary (llVISOlS, 144 Euler's fwwtion, In Factor group, 8M QuotienL group l!'ouf-group, 4fJ, 102, 118 Frobemus, 5H Galois, 120 Generntmg elomenL, 24 151
152
INTRODUCTION TO THE THEORY OF FINITE GROUPS
Group, AbelIan, 3, 126, 136-150 alternatmg, 77, 120 t£ cornmututIve, 3 cychc, 2,1" 38 dlCychc,53 dihedral, 52 dodecahedral, 93 hexahedral, 92 icosahedral, 93 mfimte,7 mtransItlVo, 82 It-ply tranSItive, 84 octahedral, 02 octlC, 52 of matnces, 9, 10, 20, 53 perfect, 105 prime power, 1341'£ prImitIVe, 80 'l.uatermon, 52 sImple, 114 soluble, II n Sylow, 127 ff symmetric, 62 tetrahedral, f} 1 transItIve, 82 HIghest common factor (H.C F ), 40 Holder, 114 HomomorphIsm, 106 Idempotent element, 7 IdentIty, 3 ImprImihve systems, 85 IInpropersubgroup, 31 Independent elements, 44 Index, 35 IntersectIOn, 40 InvarIant complex, 99 Invariant clement, 98 InvarIant snbgroup, 99 Invariants of an Abehan glOUp, 148 Invel'lle element, 3 Isomorphism, 19, 79, 106 Jordan, 78, 114 Lagrange's Theorem, 34, 140 Latin square, 13 Law of compositlOn, 1, 4, 14, 18
Least cummon multIple (L C M.), 40 if Lorentz Croup, 27 MaXImum IllVal'Iant subgroup, 11'1
Miller, 140 Moclulns, 15 MultIphcahon tablo, 11 Normal subgroup, 99 Normahzor, 97, 99 Order of an element, 21 Older of a group, 9 Partition, 71 PerIOcl, 21 Permutablo elements, 3 PermutatIOns, 62 fl. circular, 66 conJugate, 70 even, 74 intranSItIve, 83 odd, 74 reguhlr,69 SImIlar, 70 Proper subgroup, 31 QuadratlC group, see Four.group Quatermons, 53 QuotIBnt glOup, 103, 110 1'£. ReCIprocal elements, 3, II RepresentatIOn, 69 faIthful, 81 regular, 81 RotatIOns, 7 Self-conJugate element, 08 Self.conJugate subgroup, 99 Structure, 1!J Sylow, 127, 140 Transformmg element, 96 Transposition, 73 H. Type of Abelian group, 147 Unit element, 3, 5, 11 Vierergruppe, Bee Four.group
