Introduction to the Theory of Finite Groups

INTRODUCTION TO 'rUE THEORY OF FINITE GHOUPf3 UNIVERSITY MATHEMATIOAL TEXTS GlJlNERAL EDITOIlS ALillXANDER C. AITKEN,...

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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

I~

Tjl~DBRMANN,

\1\lfO.,!\.'IH ....

l~

ULIVI~H.

'IHI

1'11.1>., {Uk

l'NHhn-,tey liP

l't!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 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


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


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,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


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-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


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


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. 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


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


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


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


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 wonl5), 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).


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


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


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


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


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


(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.


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


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}.


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 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: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


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


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.


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


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


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


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


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


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


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


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


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


'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/ 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 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)=H2Sed, 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:


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-==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


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


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-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


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 [


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


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, (hO. 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.


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 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


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


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 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


(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, (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


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

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