A STUDY OP ABELXAN GROUPS

A THESIS

IN MATHEMATICS

by

John Hermon Caskey

Approved

^. * 6c,*cvCa- ^

Dean oí* the Graduate School

Texas Technolofrlcal College

May, 1955

(^"W

A STUDY OP ABELIAN GROUPS

A THESIS

IN MTHEMATICS

Submltted to the Graduate Faculty of Texas Technological College

in Partial PiOfillment of the Requireraents for the Degree of

MASTER OP SCIENCB

by

John Hermon Caskey, B, S.

Quanah, Texas

May, 1955

T5

TABLK OP CONTENTS

P a g e LIST OP TABLES • . . . . i v

Chapter

INTRODUCTION 1

I . GROUPS , 6 D e f i n l t i o n of a Gro\ap Permutation Groups Theory of Groups

I I , ABELIAN GROUPS . . . . . ik Introduct lon t o Abelian Groups fíistory of Abel Abelian Groups Theorenm

BIBLIOGRAPHY . . . • • • . • . • . . . • • • . . • * 58

i i i

LIST OP TABLES

Table Page

1. Multipllcation Table on Three Symbols • . . . 8

2. Multiplication Table for i, -i, 1, -1 . . • . 20

3« Multiplicat on Table . . . . . . . . . . . . 21

i.. Multiplication Table of a Gyclic Group Of Order Pive • • • • • • . . . . . . . . . . 25

5» Mult plicat on Table of the Abstract Group of Order Eight Containing no Element of Per od Greater Than Two. . . . . . 31

iv

INTRODUCTION

Many mathematical objects and symbols are In such

cominon use that few people conslder how these objects be-

came associated. The most commonplace of these are perhaps

the natural numbers, the integers, the rational ntunbers and

the real numbers. These symbols, together with operations

upon the symbols, are presented in groips or sets in elemen-

tary college texts. These operations are called addition,

multiplication, subtraction and division.

Although t is not necessary to justify in any do-

tail how and why these objects are presented in sets and

what the operations are and how they are defined, it is

worthwhile to reconstruct, at least partially, the climate

of opinion existent in the days when the forming and de-

fining of present day algebra 'vvas underway. It is only thus

that the tremendous work of the mathematician in those times,

when the fundamental operat ons rere not yet established,

may be appreciated. In the year 1889, only sixty-six years

ago, an Italian îTiathematÍcian, G. Peano, introduced the

Peano Postulates, îéiich led to tho development of the

algebra that we enjoy toda;/.

At this juncture, familiarity with certain basic

terms of algebra, beginning with the notion of set and

1

•lementt i s r«q\al«ite« Provided i t i s asstimed that a given

slement i s or i s not an element of a particular set» then

that set l e defined i f , given any element, i t can be deter-

xBÍned whether or not that eloment i s an element of the s e t .

Por øxassplB, the points on a plane may be considered as

elements. Then a l i n e in the plane i s defined i f , given any

point on the plane, i t can be determined whether or not that

polnt i s a point of the l i n e . I f no points can be shown to

be points of the l i n e , then there e x i s t s a null set or nul l

l ine which has no geometrical interpretation*

If the elements of a set S are also the eleraents of

a se t T> then S i s a subset of T. Associated with two se t s

S and T I s the product set SXT, which cons i s t s of a l l pairs

( s , t ) of elements of the two s e t s , where s i s an element of

S, and t i s an element of T, Thls does not , of course, im-

ply mult ipl icat ion; at t h i s point no mention has been made

of operations vípon elements. A mapping of one set into

another set i s a part icular type of correspondence between

the s e t s S and T that associates with each element of S a

unique element of T.

ûi: a oL = u means that under the raap-

ping ot of S into T, where £ i s an eleraent of S, a maps into

u, an element of T. A one-to-one mapping of a set S into a

aet T i s a mapping whereby every eleraent of T i s the unique

correspondent of an element of S, and every eleraent of T

has a unique correspondent in S,

An operat ion on the s e t S i s a mapping of S X S

i n t o S. I f ût i s a raapping of S X S i n t o S, then the element

of S corresponding t o (a ,b) of S X S i s a <y b . Por example,

each p a i r of natural numbers (a,b) has a unique correspond-

ing natural number a^+bj so ( + ) i s a mapping of N X N

(N represents the natural numbers) i n t o N and •»- i s an

operation on N. The common operations with which one dea ls

are c a l l e d addi t ion and m u l t i p l i c a t i o n and are represented

by -j- and - , r e s p e c t i v e l y . The operat ions subtract ion and

d i v i s i o n are then defined in terras of addi t ion and mul t i -

p l i c a t i o n , r e s p e c t i v e l y .

This b r i e f d i s c u s s i o n of bas ic terms, inasmuch as

i t does not extend beyond a bare statement of those terms,

cannot be considered a d e t a i l e d d e f i n i t i o n . I t I s suggested

that each term be supplemented by the "comraonsense" d e f i -

n i t i o n normally a s soc ia ted with i t . A complete and d e t a i l e d

study invo lv ing these and r e l a t e d ideas raay be found i n A

F i r s t Course i n Abstract Algebra by Richard E. Johnson.

I t has been iraplled that the elements of s e t s are

character ized by the fac t that when any two of thera are com-

b ned, another eleraent i s forraed. In other words, the e l e -

ments are subject to láws of combination v/hlch we have

c a l l e d operat ions ; t h s leads to the d e f i n i t i o n of an a l r e -

bra ic system.

An a lgebraic system c o n s i s t s of a se t of elemonts,

together with the operat ions on these e lements . I t may

k øon^idn a f l i t i te «r «n in f ln l t e ntaaber of elementa* For

•xavple^ th« potfitiire l!tt#gex»« with the operations aââition

mâ isultiplleAtion fom an aigebraie s^Btm with an in f in i te

nuBib«r of elemeiat«# Thø poaitive integez*a that are smaXler

than a partio\iIar integer together wlth the same operatlons

oonatitute an «Xgebx»aÍo s^stem with m f in i t e ntimber of eXe*

røente* I t shoi Xâ bo noted that an aXgebt^ie ø^stem i s not

neoeaaariXy oXosedl x*eXative to the operatlons, In other

vGtån$ a o bf reaå j | operation b^ where a, and h are eXezasnte

of the aXgebFaÍo a:sr^«% i s not neeessailXy an eXemnt of

the aXgebx^aie a^stn^* Xt shotdd nøw be eviéent, at Xaaat

in a bx*oad sense, why tbe Integers and naturaX numbers ond

other ©Xomeata ar® presønted in sets» The deflning of other

eonerete exaa^Xe® of aXgebraic s^stems oouXá be pursued

ÍnâeflnÍteX^j^ biit at th!s pcint It i s wise to injeet the Idea

ôf gmieraX aXgebraic syatems of nÉiich these oonorete systems

are but paz^tiouXar øxms^løB*

Om nmh gømmX type of aXgebj^c systora Is the

integraX ûemktni ® second i s the fieXd and stîXX a thiinî i s

the groi^* This, then, i s ©ufficient for purposes of def i -

nitionf attention wiXX be foctused for the remainder of th is

study ijpon the grotjp-*the AbeXian grot^ in particuXar*

The theoi^ of øpoi^s was concelved about one fetjndi»ed yeare ago to ald In solving for the roots of a páiynomÍaX, To be more procise, the probXem of determlning i f the roots of a l^ven poXynomÍaX can bo ©xpressed in terna of powers and x»oots of the coefficlants of the

poXynomÍaX was reduced to a problem in grotQ) theory, In the century since then, groiî) theoxn/' has become widely used in geometryj analysis and even in mathematical phy«ics,X

1 Richard E. Johnson, A First Course in I.odern

AXgebra (New York: Prentice-îîall, Inc, 1953T, ^* 121.

CHAPTER I

GROUPS

Definition of a Grovqp

A group is an algebraic system made up of a set of

elements and an operation o , in which the following proper-

ties hold:

(1) The olosure property holds.

(2) The associative law holds^ (3) The identity element 3 exists. Íi\.) Every eleraent a has an inverse a

a o b = an element of the system. a o (b o c) = (a o b)oc, a*»X=loa = a^ a o a"= a"' o a = 1.

During tho course of this study we shall from time

to time refer to permutation groups as examples, In fact,

this thesis is written in the light of permutation grotps,

A permutation group is by no means the only kind of group;

it is only a particular example of a group. fíow, then, can

we Justify the assertion that thls is a study of Abelian

groups in general and not Abelian permutat on groups only?

That Justification shall be made with the stateraent of

Cayley*s theorera. Inasmuch as it would be rather awkward to

present this theorem at this stage, mere reference to it

must suffice.

Permutation Grotg)S

A one«to«one mapping of a set S onto i t s e l f i s calXed

a permutation of S, The set consist ing of a l l permutations

of S I s represented by P(S) . A set of elements G of P(S)

and the product operation form a permutation group i f j

(X) G i s cXosed reXative to the product operation, (2) The assoc la t ive law holds^ Q) The ident i ty element 1 i s an element of G* (4) Evei^ element & of G Kas an inverse s."' •

I t follows iimiediately that P(S) i s a permutation

group^ Likewise jL i s a permutation grot?). Any pormutation

group contained in P(S) i s a subpemutation group of P(S) ,

Purthermore, P(S) i s considered to be a subpermutation group

of i t s e l f .

A set S of three eXements a, b , c may be considered

as example, Let P(S) be a l l poss ible pemutations of S

onto S. There are s i x such permutatlons^

ot.

* 6

a û>, = a ,

aûtj = b .

a otg = c ,

aoí , = b .

a oL - c ,

ao«- = a .

b o t . - b .

h<^z = o,

bcL3=a,

bí* = a .

\)d^=ht

h \ = c,

C A = C j

c otj = a j

c<*3 = b i

coi, = c .

c oi = a j

c ^ , - b .

, o r s, - 1 .

^ %=(abc).

f S 3 = ( a c b ) .

, ^ = ( a b ) .

^ s^ = ( a c ) .

s = ( b c ) .

8

A mtiXtipXication table of this permutation groip

( P3 } is oonstzmcted beXow,

5. S,

s s. s.

s, 5 . ^

% ^3 5. '

^ ^ s^ •

V V 4 ^

\ 4 s í

Ss S^ S ^ ^4

S* ^ V ^4

S. S, 5 , S^

5 i ^s- ^c ^^

\ ^/ s» S^

í ^a ^, S^^JI

>5- ^ ^3 s, L

The eXement at the intersection of the fourth row

and f Ifth column is s - s = s , and the element at the in-

tersection of the fifth row and the fourth column is

s s = s. 5 6

By inspection of the multiplication table, the sub*

permutation groups of P, are

s s s s s s

, s , S3 , S^ , Sy, S^ , s^ , s.

The number of d s t i n c t elements n each of these

permutat on groups and perrautation subgroups i s t l e order of

t h a t permutat ion group or subgroup. Hence, P^ i s of order

six,

Two gxwups of equal order a re s inply isomorphic i f t

(X) A one«to«one correspondence e x i s t s between the

eXements of the two groups;

(2) The product of any two eXements of one grot?)

corresponds to the product of the corresponding elements of

the other group.

Cayley's Theorem

A group of order £ can be expressed as (is Ísomo2T)hic

with) a permutation grot ) on £ symbols,

After careful consideration of the last two state-

ments, it becomes evident that reference to a group of order

£ Xikewise invoXves reference to some permutation group on

£ symbols. The original group and the permutat on group have

the same properties^ fíence, it might be said that groi?)s

in an atmosphere of permutation groups can be studied, which

is precisely what will be done in the raaterial following^

Theory of Groups

The theory of groups offered will be only a set of

definitions and theorems wldch one will need to be far.dliar

with in order to follow the arguments of the theorems on

Abelian groups, Decause proofs or explanat ons of the

theoreras and terms would entail a rather detailed and len;;thy

process, equalling—if not actually overshadowing—the ma n

body of this thesis, such proofs and terras are not offered.

xo

Befini t ions

X» Â sttbgroup, as might be expected, i s a grot:^

within a gXHít;?). The ident i ty eXeraent of a group i s a sub-

grotQ) of the grot^j the grot:qp i s also a sixbgiHSup of itseXf •

A proper subgroup of a grovp i s any subgrot?) of the grotg)

other than the gro^p ÍtseXf•

2 , I f G i s a group and H i s some subgroup of G, a

Cauchy tabXe of H may be constinicted as foXlows:

H j h , , hjj , • • • . . . , h«. H g . : h , g ^ , h ^ g , , . . . . • . , h^gj^

Hg t b, g^, \ g ^ # . . • . . . , h^g^ ,

where the g's are d i s t inc t elements of G not contained in H

or in any l ine above the l ine which contains the particular

£ being considered, Each l i n e of the Cauchy table i s a co-

set of G with respect to H,

3 . The period of an element £ i s the sraallest

pos i t ive integer n for which s''- 1.,

14.. Two [ ^gpoups^ } are said to be independent i f

they have no { I JJJQJI-I; Í in common In the case of groups

the ident i ty element w i l l , of course, be comraon to a l l

groups.

5^ The transforra of an element ^ by an elernent t i s

defined to be t £ jb

6^ A group which can be formed by raising one

part icular element, £ , success ively to £ , where £** = 1., i s

a cyc l i c group, and £ i s a generator of that group. A group

11

i s sa id to be generated by a se t of elements, provided a l l

the elements of t he group can be obtained by combining the

elements and powers of the elements of the s e t . "lien the

set c o n s i s t s of a minimtmi number of elements, we say these

elements a re genera tors of the group^

7» X^ Í î l i i ' l * where t^, h^ , and h^ are elements

of a group G, t i s said t o transform h: i n t o h^ . I f

í hí í "^ hí» îii ^^ sa id to be i nva r i an t tmder Jb A se t of

elements of a group G i s said to be i nva r i an t under an e l e -

ment t of G, i f t t ransforms them among themselves. A sub-

group fí i s said to be i nva r i an t under t t s group G i f every

element of G transforms H i n t o i t s e l f •

8. Al l the eleraents common to two or raore groups

form the c ross -cu t of the groups.

9^ The elements in to which a given eleraent of a

group i s transformed by a l l the eleraents of t h a t group form

the conjugate c l a s s of t h a t element with respect to the

group. The subgroups i n to which a given subgroup of a group

i s transformed by a l l the elements of t h a t group form the

c lass of conjugate subgroi;?)^ of the given subgroup v/ith

respect to the group.

10, The c e n t r a l of a group i s the t o t a l i t y of e l e -

ments t h a t are t h e i r ovm conjugates , t ha t i s , the eleraents

t h a t are comîmtative with every eleraent of the p-roup.

11 , I f £ and ;t a re any two eleraents of a rroup G,

then £ = £"' t "' £ t i s the cora.-riutator of £ and _t, not of ;t

X2

and £• I n o t h e r words, t s c a s t ,

X2, I f G and H a re indepondent grot?)S, and i f every

eXement of G conimutes with evex^ element of H, then a l l pos -

sÍbXe px^duots of one eXement of one group by one element of

^he o ther group form the d i r e c t product of G and H, w r i t t e n

G V H. The order of G X H i s the product of the orders of

G and H,

X3« Let H be an i n v a r i a n t subgroup of the gi^tjp G,

and Xet TT be the p a r t i t i o n of G in to se ta of eXements such

t h a t % ^ ( a, , Ha^, • , • • . , M&^^ , where a, - ! • Then l

Í s a groi?) and i s ca l l ed the quot ient grot:^ of G r e l a t i v e to

H or the f a c t o r grot:qp of H in Q* I t s order i s ^ , where

m i s the order of G and n i s the order of H.

Theorems (Baslc)

1 . The order of a subgrot?) H of a group G i s a

d iv i so r of the order of G, This theorem i s ca l l ed the

Lagrange theorem and î s the fundamental theorera of grotip

s tudy. Tho converse of the theorem i s not necessa r i ly t r u e .

2 . The period of any element of the group i s a

d iv i so r of the order of the group,

3 . A group of order £ can be ejqDressed as ( i s

isomoiphic with) a perrautation on £ symbols. This i s Cayley»s

theorem and i s the second raost important theorem in group

s tudy .

13

l^m The elements of the centra l of a grotjp form a

grotqp •

The Xast s e c t i o n per ta in ing t o grotaps i s a b r i e f

presen ta t ion of the Sylow theory including the th ird most

iiHportant theorem as regards group study,

Sylow Theory

By Lagrange's theorem we know that the order of a

subgrot:^ of a grovp G must d iv ide the order of G» I f the

order of a grot;^ G conta ins as a faotor the prime ntmiber p ,

then G contains at l e a s t one subgroup of order p . This i s

known as Cauchy*s Lemma,

! • I f a prime ntmiber p d iv ides the order of G,

then G contains some elements of period p .

2 . I f the order of a group G contains p*" and no

greater power of p , then G contains subgroups of orders p ,

where O ^ s ^ m . This i s Sylow»s theorem. Al so , i f the order

of G conta ns as a f a c t o r p** and no higher powers of p ,

then a l l the subgroups of order p*^ are ca l l ed Sylow sub-

groups.

3 . Every subgroup of order p* , s < ra, i s contained

i n at l e a s t one Sylow subgroup of order p*" in G.

if.. A l l Sylow subgroups of order p"" are conjugate .

5 . The ni; mber of Sylow subgroups of order p*" i s

kp •«- 1 , where k i s a p o s i t i v e i n t e g e r .

CHAPTER II

ABELIAN GROUPS

ntroduction to Abelian Groups

A group i s defined, br i e f l y , as a set of elements

with a ffiode of combination cal led mult ipl ication having the

following propertiesj

(X) The closure property holds. (2) The associative law holds. (3) The identity element exists. (4) Evory element of the set has an inverse.

If, however, a fifth property, the commutative law,

hoXds for every pair of eXements of the group such that

st = ts , where s and t are any

elements of the grotjp then the group is a special kind of

grotjp and is called an Abelian (from Niels Henrik Abel)

group. A brief history of Abel follows this introductlon,

Two elements are said to commute (or to be commutative, or

permutable) if st = ts .

Biography of Abel

Niels Henrik Abel was born on the island of Fino,

off the coast of Norway, on August 5, 1802, and died on

Aprll 6, 1829. He carae from a poor family, and his life

was full of misery, family worries, and unmerited neglect by

X5

the poXitÍoaX and soientlflc raAsters of that day, Bven hia

briXXiant mathematicaX disooveries reoeived no recognltion

tantiX the Xast year of his Xlfe.

AbeX received his earXiest education at the hands

of his father, a thoroughXy talented man. In X8X5, the yotang

mathematieian, ahose genius had to await posthumotis recog»

nition, entered the Cathedral School of Christiania, at

which institution he evinced littXe interest in his studies

for two or three years. The appointment of Bemt îsichael

HoXmboe as a teacher, however, changed all of this. Under

the infXuence of and directed by HoXmboe, AbeX was stimulated

into an intellectual activity of whioh his prior perforraance

had given no promise. At the instigation of his mentor, he

avidly perused the works of Lacroix Prancoetar, Poisson, Gauss,

Gamier, and particularly Lagrange.

In 1820, at the age of eighteen, Abel completed a

one htandred and ninety-two page notebook entitled "Exercises

in Higher Mathematics by Niels Henrik Abel," a work which

clearly reveals its author to have been wholly conversant in

the theory of functions and particularly interested in the

theory of equations. It was at tliis time, only six months

before his examination for admission to the University of

Christiania, where he hoped to get an appointment because

of his aptitude for mathematics, that Abel occupied himself

with the solution of the general equation of the fifth de-

gree. In this same year, incidentally, Abel»s economic

16

s i t u a t i o n became increasingXy corapXicated, h i s f a t h e r having

d ied and Xeft hira to stapport h i s mother and s i x yotmger

chiXdren.

Some months before h i s examinatlon f o r entrance to

the U n i v e r s i t y , AbeX presented h i s soXutÍon of the generaX

qt i int ic t o the Royai Society of Sciences of Denmark. The

soXution was examined by Carl Degen, Professor of Matheraatics

at the Unlvers i ty , who did not tmderstand the true nattire of

the probXera. His repXy came on May 2 1 , 1821, and, while ex*

p r e s s i n g great admiratîon for AbeX, he showed scept ic i sm and

caut ion without po int ing out d e f e c t s . But, before rece iv ing

Degen^s repXy, the yotmger man himseXf had discovered the

defec t which completely v i t i a t e d h i s s o l u t i o n . Hence, i t

was Abel, at the age of e ighteen , who proved that the general

equation of the f i f t h degree could not be so lved . However,

h i s theory was so complicated that i t was not widely accepted

at that t ime . Then, t o o , h i s b r i l l i a n c e hindered him to the

extent that he could not present h i s t h e o r i e s in a manner

or form which the average student could f o l l o w .

Abel entered the Univers i ty in Ju ly , 1821, and wi th in

two years he had care fu l l y studied a l l the woiks i n the

varlous l i b r a r i e s at Chris t iania and had at ta ined a very high

reputat ion at the University as a reraarkable mathematician.

During t h i s tirae he was not without t rouble ; he had entered

the Univers i ty a very poor man, and only through a fund

""inni

X7

maintained by friends and coXXeagues had he been abXe to re-

main there.

It was dtiring the year X823 that AbeX became con-

vinced that none of his teaohere could aid him any furtherj

consequentXy, he conceived the idea of making a Grand Totir

of the Continent in order that he might meet such eminent

men as LapXaoe, Gauss, Poisson, and Legendre, On March 23,

X823, in an atteB?)t to proctire a subsidy from the University

for his traveX, he submitted a memoir, **A general exposition

of the possibiXity of integrating all kinds of differentiaXs,"

to the proper authorities, Some nine months Xater, in De-

cember of 1823, his memoir was retumed aXong with a recom-

mendation for such a subsidy as he desired, His hope, of

coiarse, was to obtain in Europe the recognition he deserved

with the paper on the fifth degree eqtiation serving as a

passport to the acknowledged great mathematicians. Nor was

his hope withotit foundation in merit; dtiring his two years

at the University he had written fotir major papers and a

dozen and a half minor ones on various phases of mathe-

matics*

Abel»s totir, his drive for reco^ition, was a dismal

failure. He did manage, however, to contribute several

papers on differential and integral calculus, algebra, and

elXiptic functions to a matheraatical Journal while so en-

gaged. But in the end he was forced to retum to liis

univorsity without either friends or ftmds. The record of

18

h i s bx4ef span of Xife, and work may be terminated with the

nota t ion tha t on Jantaary 6, X829, he became iXX of bron-

c h i t i s , and he died during the raoming of ApriX 6, X829.

But such a sketch as i s here presented i s incomplete without

something more being said of the general equation of the

f i f t h degree. I t i s only thus tha t the fu l l s ignificance of

Abel 's work can be rea l ized .

Special cases of the quadratic equation, as well as

particuXar cubic and even some quart ic equations are known

to have been solved as ear ly as 1950 B.C. Quadratic equations

were solved arÍthmeticalXy by the Egyptians, geometrically

by Euclid, and a lgebraical ly by the Hindus. fíowever, Simon

Stevin (1585) seems to have been the f i r s t t o use a single

formula for a l l quadra t ics . The solut ion of the general

cubic equation was published in I5if5 by Girolamo Cardano

(1501-1576), an I t a l i a n mathematician. Another I t a l i a n ,

Nlcolo Tar tag l ia , claiiiied that he had discovered the solutlon

which Cardano published and that he had i riparted the knowl-

edge to Cardano only af ter f i r s t pledgins the l a t t e r to

secrecy. Re contended furthermore that Cardano subsequently

divulged t M s mater ia l without perraission. Though t h i s con-

troversy was never s a t i s f ac to r î l y s e t t l e d , the solution t o -

day bears Cardano's name. Gardano also published, with due

acknowledgment of the proper authorship, Luigl Ferrar i»s

solution of the general quar t ic equation. Slnce Ferrar i»s

X9

soXtxtion of the quartic equation invoXved the soXution of

the cubio eqtuition, it was a naturaX assumption that the

generaX qtiintic equation could be soXved and that the soXu-

tion wouXd aXso invoXve the solution of the quartic equation.

With this assuffiption in mind such eminent mathe-

maticians as Porro, TartagXia, Cardano, and Perrari con-

sidered the problem in the sixteenth century. During the

eigíiteenth centtiry further efforts were put forward by

Beíout, Lagrange, Vandermonde, and Malfatti. Althotigh their

results were rich In valtiable research, they had little

success. Consequently, the solution of the general equation

of the fifth degree appeared to be an iuposs b lity. This,

then, was the light in which mathemat cians regarded the

problem at the time Abel began work on it.

Abelian Groups

Vilhen £ and t are elements of a grotjp G, we have, in

general, (st) *• ^ t''\ but if the group G is

an Abelian group, then (st)'' = s" t" . Also, if the group G

is an Abelian group, then 3*" " = t** s' .

It is evident, after a b t of consideration, that

the study of Abelian groups is more systeraatic and coraplete

than the study of groups in general. In the study of Abelian

groups when one knov/s the product of two elements £ and t ,

then one also knows the product of t_ and £. In the general

group study, however, this is not generally the case. Hence,

20

a survey of all possibXe abstract AbeXian groups can be

made. Thls cannot be done with groups in generaX.

An exa pXe of a finite Abelian grotjp is the group

containing the four elements 1, -1, i, -i, or G - X,-l,i,-i

A consideration of the following multiplication table of

this group will reveal that it is indeed an Abelian group.

/

- /

Â

/

/

- /

^

- ^

- /

- /

/

-Á,

0

JL

JU

r

-/

1

-Åj

ê

JU

1

-J

21

The Abelian groti$), the mtiltipXicatÍon table of which appears

below, i s freqtientXy referred to as an example of the

theorems.

MultipXication Table"^

t

s

z.

s

s'

t st

s \

s'í •

X

X

s

S

3 S

t

st

s-t

s't

S

S

í

s 3

S

1

st

s't

s't

t

s

2

s

3

s

1

5

s't

s't

t

st 1

s

s

l

s

s

s'/

í

s í

s't

t

t

st

s*í

s'í

X

s

s

3 5

st

st

^t

s't

t

s

s

3 S

1

^t

s^t

s't

t

st

s

s'

1

s

s ' .

s^t

t

st

s V

s

I

s 2.

s

s = s

(s^r (s^ ) '

t"

(st)"

(sS)' (s't)"'

= s

- s

= t

= s"*t

- s^t

- st

Mource: Walter Ledermann, Introduction to the Theory of Finite Groups (Kew York: Interscience Publ shers, tnc . , 1W9T

This group i s generated by s = t = jL. The sub-

groups of t h i s group are as fo l lows:

S. •= l , s , s s

S = X, s^, s t , s ' t

S j = X , S

S = X, t

S = 1, s*t

22

Theorems

Theorem 1

^» S t a t e m e n t ; A l l c y c l i c groups a r e Abe l i an .

2 . C o n d i t i o n s ; Let G be a c y c l i c g roup , of o rder g , which

i s gene ra t ed by some element of G, say £ ( s = 1 ) .

3* Proof; S ince t h e group G i s c y c l i c , ever: element of G

Í s a power of £ and raa^/ be r e p r e s e n t e d as such . Hence, we

a r e a t t e î i ? ) t ing t o prove t h a t

(1) s"s = 5%-", where a and b are

less tHan g.~"

If the left side of (1) is expressed as

* — -"V.

r s s s ...... s 8 s ...... then by the

associative law

S S S . . . S S S . . . ^ s s s . . . s s s . . .

or ^ B° = s" s* , and the proof is complete.

Example; The group conta ning the fotæ elements 1,-1,:' ,-i

is cyclic.

(-1)' = -1 ( )' = i (-i)' = -i

(-1)' = 1 (i)'—1 (-i)' ' -1

(-1)' = -1 (i)'--i (-i)' - i

(-1)'' = 1 (i)''= 1 (-i)" • 1 .

23

'•' t ^•:r,.îi'.. •• C o r o X X a r y '^ r^^

^* Stateawiti Any ø?o%xp whioh hae no pvop^r subgrot:^ i s ,

AbeXian»

^* Z222Í* Bvepy eXement (eaÊcept the identity) of a grot?)

0 generates a subgpot:q? of G. Henoe, i f there are no proper

atåbgXH>t s of G, then there must be no eXeraent of G of period

Xess than the order of the grotap. I t foXlows that every

eXement (except the identity) must generate the grot^.

fíenee, the group i s cyc l i c , and by the theorem i t must be

AbeXian*

Theorem 2

^* ^tatement; AXX subgroups of an Abelian grot^ are

invariant.

2» Conditions; Let G be an Abelian group, and let H be a

subgroiip of G.

3« Froof: A subgroup fí of an åbelian grot^ G is Abelian.

Hence, H is certainXy transformed into itself by every -.1

element of G; t Ht = R

and Ht - tH.

It follows that H is an invariant subgroup of G by defi-

niti)n.

Example; Consider the Âbelian group on page 21.

(s^ )S. s* = l,s,s*,s'=S, s"'s^ s =l,s , sí, sí = Si

(s^ )" S. s' = l,s,s*,s' = S. (s^)S^ s = l,s*, st, s*é=Sj

s" S, s =« l,s,s*,s*= S, (sMs^ s =l>s^, sí, s't = S

^

t s.t •t^' s, (»t)

a t

•'t r«

s

t

at

sH

s't

S,

s.

s. * l s.

s. - t

- I

s.

Sa

s fs^ » ) ^ r

8 ) S^

t )"'S^

8t)" ' S^

» t )

•H)

S)

s*)

a' )

t )

8t )

s't)

s't)

i i

^Xf8 | fa j | 8 - S i

~jL|i8y8 j | 8 - o ,

' X f 8 | 8 ^ , 8 ' S , (S t

- X , S , 8 i |S~ 2S, \ 8 V

- X ,S ' S^

= X ,S* - S ,

= X^S* = S j

= X , s * » S3

~ I f S ~ S3

" JL, 8 — Oj

~ l^ 8 = Sg

s ) = XtS t

8 ) ~ X ,S t

s ) = Xj|S t

t ) = 1,& t

St / ' X , S w

-- S,-

= s.

= 8

s"t)"s^. ( s * t ) a , s t =

S^-

V

s t í ' s ^ ( s ' t ) - l , s t - S^

(8

( 8 *

( s '

(t )

(st

(s*t

(sH

t S^ t = Xj S , 8 t , 8 t = S^

St)"s . (8t)'X^8*,8t, s ' t = S

S.

S.

Ski

s. . 1

- I

- f

a,

s,

B.

S.

8 t ) = X , 8 * , 8 t t s ' t = S,

8 t ) = Xf S * , S t > s ' t

s) = X,t =S^

s*) = X,t = S^

s / •= l , t - S„

t ) » x , t = S

• fcL'

)= x,t = s S * t ) - l , t = S^

s ' t ) - - l , t = S^

s.

. • • • i ' . « ; • • - <

Hence, a l l the subgrot^^s of t h i s group are invar ian t .

25

Theoxwn 3

^* j tatementt A gx*ot:i> of priaie oráer is AbeXÍan*

^* Conditionai Let 0 be a gx»oup of order p^ where p is a

prime nuaber*

3» ££22£í Sinoe the order of the group is p (a prime ntimber),

the group has onXy two aubgrot sf one of order one (the

identity eXement) and one of order p (the gvovp itseXf). If

£ is any element of ø other than the idmtity, then its

period must be p (since it wiXX generate a subgroup of G).

Hence, by definition, the group ia cycXic»

^* CoxioXt SÍont By theorem X, every eycXic grotqp is AbeXian.

Since It has been shown that aXX groups of prlme order are

eyeXic, then it follows that they must be Abelian and the

proof is completc.

Sxa le; The muXtipXÍcation tabXe of the group generated by

an eXement £ of period five is given below. The theorem can

be verified by a study of this tabXe.

/ s s s z^

; / 5 5 S 5

S S S S 5 i

s' s' »' í s »"

5 5^ I S 5 5 . . I I •

ss* 3

ss

ss**

z 3

s s 2. ^

a s i -4

s s

:r

=

r

• = .

~

=

3 S

3

X

X

s

s

-

=

=

s:

—

s

2

s s 3

8 8

s s 3 i

S 8

8 8

•«1 3

s s

26

Theorera \^

•^* Statement; The central of any group is an Abelian group.

2. Proof; By deflnition, the eentral of a group is the

totality of self-conjugate elements of the group. In other

words, an element of the central Is transformed into itself

by every other element of the group. If £ and t are any two

eleraents of the central then t * st = s

or st - ts , and the central

is an Abelian group.

3» Conclusion; It follows that if a group is Abelian, the

central of that group is the group itself. The central of

a grotap may be thought of as the "Abellan part" of the

group.

Exaií^le; Conslder the non-Abel an subgroup (of order eight)

of the symmetrlc group on four symbols which consists of

the followlng elements;

s. - 1, s = (12) Olj.), s, - (13) {2k), s^ =: (Ik) (23),

V = fl3), s^ = i2k), s, '(123i|), s, - (lii-32).

s, (s

s/ (s.

(3. s -I

s,- (s.

s. = s.

s_ = s.

s, ^ s.

= s

= s

. I

ís.

s/' (s.

s,- (s

s;' (s,

^' (s

s = s

S. =: S,

s, = s.

s.. = s.

S, s S.

s,*' (s^)s,

s."' (Sí.)s,

s/' (s^^s^

s*-

= s.

^ s.

~ s

8,

s

s I

s - I

7

s 8

( « j

(«5

( » .

<«,

(3,

(s.

•, '3

• , = ^

a "s s

s^ = s 5 3

S = s.

s. 8.

27

8,'' (sjs, = a,

s;' (s )s = Sg

s;' (Sa)s, = s,

K (»s)^ = 7

Hence, the central of this group is the

Abelian subgroup consisting of s and s

s = s

Theorem 5

^* Statement; In an Abelian grot^) all commutators are

equal to the identity element.

2. Proof; If £ and t are any two elements of a grotip G,

then the commutator of £ and t is defined as

c = s''t" st.

However, if G is an Abel an groi;^, then

st = ts

and c = s~'t"' ts

c ~ s ' Is

c 1., the identity element.

Hence, all comrautators of an Abelian group are equal to the

identity element, and the proof of the theorem is complete.

Exa ple; Consider the ;\belian group on page 21. It is

obvious that if the identity is one of the two elements con-

sidered, the commutator is the identity element. The com-

mutators of several pairs of elements appear below.

s (t)s (t) = 1 z 2 3

s ss s ~ 1 ? 2- *• _

s s ss - 1

28

(s't^s'íst) = 1 s*ts'(t) = 1 s'sss' = 1

sía^t^s'(s%) = 1 s^^s't^s^^st) =1 s'ts(t) -1

s(st)s^ (a't) = 1 s^(s''t)s''(s*t) ^l s'(s't)s(st) =1

s*(st) s*(s't) =1 s'^s^t^s^s't) =1 3 9

3 (St)s(s t) = 1_

It can be shown in like manner that all commutators are

equal to the identity.

Theorem 6

^* Statement; If the order n of an Abelian groi;?) G is

divisible by an integer ja, then G contains a subgroup of

order a.

^* Conditions; Let the order n of an Abelian group G be

expressed as (i) n = PT'P^* ••• pj^" » where p, ,P^,»«.,P^ are

prime ntimbers and m. ,m^,...,m^ are positive integers;

(ii) n^p*^ , where p is a prime number and m is

a positive integer.

3. Proof; If n-p*^, the divlsors of n are of the form

a = p*** (m. ^m), and there is at least one subgroup correspond-

ing to each £ (page 13) •

If n = p 'p*"* ...p* **, then a divisor of n can be of

any of the following forms;

(a) a^p^ (i-^ 1,2,3,...k). Then G contains ele-

ments of period p; hence, G contains subgroups ôf order p

(page 13).

(b) a^pr** (i 1,2,3,...,k). Then G contains at

29

l e a s t one subgroup of order p. ' (page 13) .

(c) a=p . •' ( i , j , = 1 , 2 , 3 , . . . , k ) and (m . <m).

Then G contains at least one subgroup of order p. ' (page X3).

(d) a=p7' p^' where p- ^ p^(i,h = 1,2,.. .,k) and

m- m^ and m, <. m^ (J,l = 1,2,.. .,k). In (c) we have shown

that there are subgroups of G of order p*^* and pf^ . The

product group of two of these subgroups will be a group

(subgrot; of G) of order a= p!"'' pj' .

k* Conclusion; All possible divisors of n have been con-

sidered, and it has been shown that there is at least one

subgroup of G of the order equal to each of these divisors.

iíence, the proof is complete. It is to be noted that this

theorem is the converse of the Lagrange theorem as ren;ards

Abelian groups, and although it is not necessarily true for

any :roup, it is true for Abelian groups.

Example; Consider the .Ahelian grovip on page 21. It is seen

that there are subgroups of order 1, 2, and I)., #iich are the

only divisors of eight.

Theorem 7

•* ^^tatement; A cyclic Abelian group G of order n contains

at least one eleraent of period n and more than one element

of period n if n 2.

2. Proof; It follows frora the definition of a cyclic £;roup

that the Abelian group G raust contain at least one eler'ient

of period n. Then s* (0 < ra <n) is alvays of period n if ra

30

and n a re r e l a t i v e l y pr ime. I f n>2, then the re i s always

at l e a s t one i n t ege r m, such tha t n>m >0 and n and m are

r e l a t i v e l y pr ime. Hence, i f n >2, G contains raore than one

element of per iod n .

k» Conclusions; I t follows tha t the generat ing element of

a group of t h l s nature i s not uniquely determined, fo r I f m

i s prime t o n , then s may be taken as the generat ing e l e -

mont of the group. However, a l l cyc l ic groups of t he same

o rde r are isomorphic, and there i s only one abs t r ac t cyc l ic

group of any o rde r .

Example; The cyc l i c group of order four on page 21 i s gen-

era ted by e i t h e r the element i or the element - i .

Corol lary

^* Stateraent; I f the order n of an Abelian group G i s a

prime ntmíber p , the number of eleraents of period p i s equal

t o p - 1 .

2 . Proof; Since p i s a prime number, the group G contains

only elements of per iods one and p . There i s only one e l e -

ment of per iod one, t ha t being the i d e n t i t y element. Hence,

the reraaining (p-1) eleraents of G raust be of period p , and

the proof i s coraplete.

Example; Considerat ion of the raultiplication tab le on pare

2^ bears out the c o r o l l a r y .

31

Theorem 8

^* Statement; If a grotip G contains no elements of period

greater than two, then G is an Abelian group.

^* Z££2£î All the elements of G are of period one or two.

The only element of period one is the identlty element.

Baoh element of period two generates a group of order two,

each of which is Abelian. The direct product of two Abelian

groiaps having only the identity element in common is an

Abelian group. If successive direct proc'ucts are taken, the

group G will eventually be produced, and it will be Abelian

since it Is the direct product of two Abelian groups.

Exaraple; The multiplication table of the abstract group of

order eight, wh ch contains no element of period greater

than two, appears below.

1

.5

t LL

U

SU,

u sZa.

t

i

s

t

u-

%t

iU.

iu.

stu

s

s

1

st

5«.

t

u.

•ituu

tu.

t t

%t

1

tu.

s

itiu

u.

SíA.

u.

u.

su.

tu.

1

itu>

s

t u

st

u t

s

stuu

l

tu.

Su

u.

SA.

Su.

ÍU

itu.

s

ÍM.

í

st

t

Íu

tu.

Stu

u.

t Su.

st

1

s

Stu.

SJUU

tu.

Su.

u ou

t

s

1

Ivalter Ledermann, ntroduction to the Theory of Pinite Groups (New York: Interscience P Elications, Tnc.,

32 Theorem 9

^» Statementf There are two grotqss of order four, both of

them AbeXian,

^* Oo^^^ itionsg Let G be a gx»ot of order g = l|.»

3» ?^ott I f g » kê then the eXements of the group, other

than the ident i ty eXement, mtist be of period two or fotir.

I f G contains an eXement of period fotir, then that eXes^nt

generatea the gx tap and G i s then a oycXic group. By

theorem X, every cyoXÍc grotip i s an AbeXian group. I f every

eXemimt of G, other than the ident i ty eXement, i s of period

two, then G i s AbeXian (theorem 8) and the proof ia complete»

k* ConoXuaiont The Xatter group i s known as the fotara

group. Since there are no other pos s ib lX i t i e s , v/e concXude

that any group of order four i s isomorphic with one of the

two grot:gps abovô»

BxampXet The AbeXÍan group on page 22 i s a cycXÍc gtovtp of

order four, The subgroup of the symmetric group (on four

symbols), which cons is ts of the fotjr elements s. = 1, s^ =

(X2) (3k)$ S3 = {X3) Í2k), s = (Xi|) (23) , i s a grot^) con-

taining onXy elements of period one or two.

Theorem XO

^* Statementf There are two abstract grot^^s of order s i x ,

one ^belian and one non-AbolÍan.

2 . Conditions; Lot G be an abstract group of order s i x .

3* Proof; By theorems (basic) 1 and 2 , the group C can

33

contaln onXy eXements of periods one, two, three, and/or

six,

(a) If the grotjp oontains an element of per od six,

then the giH>up is cycXic and hence must be Abelian.

(b) If the grot5) contains no elements of period

six, then the period of every element (other than the iden-

tity) must be either two or three. Since the order of G is

not a power of two, not all of the elements can be of period

two. Hence, there must be at least one element of perlod

three so that s, s*, s =1 are three distinct ele-

ments of G. Let t be a fourth distinct eløment of G; in

other words, £ and t are independent. The six distinct ele-

ments of G are i, s, s*, t, st, sH because,

(1) if st^ 1 =s^, then t =s"' s'= s* (a contradiction);

(2) Íf st =s, then t - s"' s = 1 (a contradiction) j

(3) if st = s , then t = s"" s^=s, (a contradiction).

Slmilarily, it can be shown that s t is different from the

other five elements.

Now we raust show that the set of elements (1) forms

a non-Abelian group. Since the set of eleraents (1) is a

grot^, the closure property must be satisf ed. In particu-

Xar t raust be equal to one of the six eleraents (1). If 2. 2 -•

(1) st = t , then s = t t , a contradiction. If

(2) s^t't^, then s* = t''t' , a contradiction. Hence,

t must be equal to one of the first three eleraents of the

set; (a) t^ = s, or (b) t = s* , or (c) t = s'= 1.

yk f ^ ^ In thefirsTtwo c"ãses t must be of period thrêê |

(sinoe It is evidentXy not of period two), or t = X. How

Xet U8 assume that (a) t*" = s, then t = st, a contradictiow.

Now Xet us assume that

(b) t = s , then t = s t, a contra-

diction. Hence, by the process of eXimÍnation,

Now we shalX consider the eiement ts, which must be

equaX to one of the eXements (1).

If ts =t, then s = 3., a oontradiction.

If ts *s, then t = ss'' , a contradiction.

If ts = s^, then t = s's"' , a contradictlon.

The only remaining possibilities are (a) ts =st,

or (b) ts = s^t. If ts = st, then G is Abelian and

(st) = s t = s =* 3.

and (st) = a^ i = t * ;*

Hence, the element st would have to be of order six,

a contradiction. By the process of elimination

ts = s''t^st, so the group is

non^Abeli an.

k. Conclusion; We have consldered the case in which the

group contained an eleraent of order six and the case in

which the group contained only eleraents of period three and

two. It was also shown that the group must contain elements

of period other than two. Hence, we have considered all of

the possibilities and have fotind only two abstract grot?)S of

35 order a ix t one AbeXian and one non-AbeXian.

Theorem XX

^* Statementt There are f i v e abs t r ac t grot:q?s of order

e i g h t , t h r e e AbeXian and two non*AbeXian.

^* Po"^<^itions8 Let 0 be an a b s t r a c t groiip of order e i g h t .

3» Proof; Each eiement of any gvovíp of order e ight (ex-

cep t ing t h e i d e n t i t y ) i s of per iod two, fotir, or e i g h t .

Hence, we have t h e foXXowÍng combinations of eXements which

might appear i n a group;

(a) EXements of per iod e i ^ t i n comb na t ion with o t h e r s ;

(b) EXements of poriod two and fotir but no eXements

of pe r iod e i g h t ;

(c) EXements of per iod two a lône ;

(d) Elements of per iod four a lone .

Let us consider these four p o s s i b i l i t i e s i n o rde r .

(a) I f the grotip G contains an element of per iod

e i g ^ t , then t h i s element generates a group of order e i g h t .

Since t h e r e can be only one abs t r ac t cyc l ic group of any

o rde r , t h i s grotip w i l l be the only grotip of order eight

which contains an element of per od e i g h t . In t h i s case 8

G = Ca , s = 1 . Also G is Abelian since every cyclic group

is Abelian.

(b) Let us consider a spec ia l case of t h i s corabination.

Let the element s ( s^ =" 1) and t ( t = 1 ) generate cyc l i c

subgroups which are independent . The c ross product of these

36

eyeXÍo 8ubgrot4>8~wi XI produce an AbeXian group"~of ô'rde'r

e iøat because the cross product of AbeXian grotips i s an

AbeXian grotip. In t h i s case G =C^X C ; where s = t "* jL,

8t = t s .

(c) I f G contains only elements of period two, then G

i s AbeXÍan (theorem 8 ) . In th i s case G = C X C X C ; where

s = t = u = X, and s t = t s , tu = u t , su = us . We have now

considered compXeteXy (a) and (c) and a speciaX case of (b) .

In doing so we have obtained threo abstract Abelian groups

of order e ight . Any other abstract group of order oight

must have an element of period four; in other words, an

element of the f orm s^ = j . . The remaining e l ^ e n t s of t h i s

group must be of period two or four. The element (s) of

period four generates the group

s , s^, s ' , s "= 1* I f t i s any eleraent of

the group not contalned in (1)—in other words, i f t i s

chosen independently of e—then the eight elements of G may

be written in the form

(2) s'' = 3., s , s^, s , t , s t , s'^t, s t .

Since (2) i s a group, the closure property must hold; in

other words, t^ and t s raust each be equal to one of the e l e -

ments of (2 ) . I t cannot be equal to one of the las t three

because i t was chosen independently of £ . If t = s , then

t i s not of period four or two, a contradiction. I f

t^= s \ then t i s not of period four or two, a contradic-

t i o n . Hence, t i s equal to neither £ or £ , and by the

37 -Z . 4 *»

prooeaa of eXlwination t = s , or t = s = 3.. P i r s t , we

wiXX aastaiie that t " !• As previousXy s ta ted , t s must aXso

be equaX t o one of the eXements ( 2 ) . In part ictaar , t s naist

h% eqtial to one of the laat three elements, since t and £

are independwit. I f t s » s t and t* = l, B"^ 1, we have

the AbeXian group (b) . I f t s = s*t , then s = t ' 8* t and

8** ( t " s * t ) { t ' s * t ) - t " s^t = t " t = 3., a contradict ion.

Hence, t s = s ' t , or s t s t = s^'tt

(st) = a t =3. . Hence, th i s group can

be defined by the re lat ions s** = ^ , t*" = 3., (st)*= 3.. Now

we wiXl assume that t* = s^, which means that both t. and £

are of period fotir. Again t s must be equaX to one of the 2. 2

Xaat three eXements (2). If ts - s t, then ts - t t and 3 Z X

t s = t , which would imply that s = t = s , an i n ^ o s s i b i l i t y .

I f t s = s t , the grot?) i s Abelian and of type (b). Hence,

t s = s t and the group may be defined by the re lat ions

s = jL, s = b , t s =- s t .

J±, ConcXusion; We have considered all of the posslbilities

and found only five groups; three Abelian and two non-Abelian.

Theorem 12

-* Statement; Every Abelian group is the direct product of

its Sylow subgroups.

2* Conditions; Let G be an Abellan group of order

g =î p* 'p"'* P "" , where the p«s are distinct prirae

38

n«mb.r« and the exponenta «re p o . l t l v . I n t . g e r s . «

î* Prooyt Since aXX subgroups of an AbeXian grot:?) are

invariant, there Is exactXy one SyXow stibgrotip correspond-

ing to each pxdme that is a factor of the order of the

AbeXian group (page XJ). Now, since the order of the

AbeXian group is g = p*' p^* .....p^* , then G has r SyXow

8Ubgx*ot2ps H,, H,....., H , whose orders are p7* » p"*" >

• ••••» P*"* 9 respectiveXy (page I3). The orders of these

Sylow subgrotqps are reXatÍveXy prime since each is a power

of a prime ntimber. Hence, the Sylow subgroups contain ele-

ments of periods which are relativeXy prime (with respect

to the giwups and not within the groups). Hence, the SyXow

subgi*otips can have no eXement in coramon except the identity.

Therefore, the grot )

H,X H X......X H^ is of order

g = p^' p**%..ep^'^ and i s i d e n t i c a l with G or

G = H, X H X X H, , which i s the d i r e c t p r o -

duct of the Sylow subgroups.

1|., Conclusions; This theorem iraplies t h a t a necessary and

s u f f i c i s n t condi t ion t h a t two Abellan groups are siraply

isomorphic i s t h a t t h e i r Sylow subgroups are siirply isomor-

p h i c . Henoe, the study of Abelian groups i s reduced to the

study of such groi^s whose orders are powers of a s ingle

prime ntimber.

39

'" Theorrø~13

^* Statement; I n an Abelian grot^ the Sylow subgroup of

order p c o n s i s t s of a l l the elements of the group whose

period i s a power of p .

^» Qonditionst Let p be one of the prime fac to r s of t he

order of the Abelian group G, and l e t H be the correspond-

ing Sylow subgroiq? of order p*** . Let R be the t o t a l i t y of

eXements of G whose per iod i s a power of p .

3« Pyooî; We s h a l l now show t h a t R = E. I f s and t are

any two eXements of the se t R and i f b i s a power of p , then

( s t ) = s t . I f ^ i s a suf f ic ien txy

Xarge power of p , then s**= 3. and t ' ' = 1, and since

( s t ) = s**t , then ( s t ) = jL.

Now, appXying the d e f i n i t i o n of a group to R;

X. The c losure proper ty holds because; I f the prod-

uct s t were not an element of period one or p'^(m = 1 , 2 , . . . ,

a ) , then R must contain some elements of per iod not equal to

1, p , or p'"(m = l , 2 , , . . , a ) , which con t rad ic t s the raanner i n

which R was chosen.

2 . The assoc a t i ve law holds .

3 . As was shown prev ious ly , the se t contains the

i d e n t i t y element.

if. Every eleraent has an i n v e r s e , a l so shown p rev ious ly .

Hence, R i s a group. Since the period of any element of

the Sylow subgroup H s a power of p then

(1) H Í s contained i n R.

1 .0

We sha l l now show that the opposite reXation i s t rue , nameXy

that R i s contained in fí. Let us suppose s to be an eXe-

ment of R not contained in H. Then the prodtict HR would be

a grottp whose order i s a power of p . This i s a contradic-

t i o n , for H i s the Sylow subgroup corresponding to p*-a

grotap whose order i s the greatest power of p . Rence,

(2) R i s contained in H.

The onXy poss ib le way that (1) and (2) can be s a t i s f i e d i s

f or R = H.

The proof i s now conqplete.

Theorem Xlf

• * Statement; Every AbeXÍan group of order p*", where p i s

a prime ntamber and m i s a pos i t ive integer , i s the direct

p i ^ u c t of independent cyc l i c grotEps.

2 . Conditions; Let G be an Abelian grot^) of order p*",

where p i s a prime ntanber and m i s a pos i t ive integer . We

sha l l consider the two poss ible cases;

(a) G i s a cyc l i c grotap;

(b) G i s a non-cyclic group.

3 . Proof; (a) I f G i s c y c l i c , then there ex i s t s in G an

element of period p* which generates the group. I f £ i s

such an element, then G = S = { s ) and ther i s nothing

further to prove.

(b) I f G i s non-cyc l ic , then there i s no element of

G which w i l l generate G. Let s, be an element of greatest

per iôd íp"*') i n G7~~The subgrotap of G generated by £ , i s

then

S. -' { s . } .

How l e t 8j be any elwnent of G which i s not an element of S, •

To provide for some senfcXance of order i t i s wise to choose

as s^ an eXement of greates t period (p*"*, where m ^ m , < m ) .

The aubgroi?) of G generated by s^ i s S^ = {s») . The prod-

uct grotap, S,S^ , i s certaÍnXy a subgrotjp of G. I t i s now

our proposaX t o prove that there can be no dupl ica t ion of

elements i n S, and S^ (excluding the i d e n t i t y e lement) .

To prove t h i s , we s h a l l assurae that there can be dupl i ca t ions

and arr ive at an absurdi ty .

To that end, we s h a l l assume that

(s, ) = . (s^) , where a < p ', and b<'p*"*'.

I t foXXows that the order of S,S^ I s then l e s s than p

(the order i f there are no dupl i ca t ions ) but tt i l l must be

a power of p . Suppose the ordor of S,S^ i s p* ' where

m <m<m. ^m. Let us now d iv ide S ^ i n t o two s e t s of e lements ,

the di;iplications and the non-dupl icat ons . Yîe s h a l l ex-

clude the i d e n t i t y element from each of these s e t s . Ve

s h a l l now prove that none of the non-duplicated elements can

be generated by one of the duplicated' e lements . I f there

were such an eleraent, then i t would be generated in S, and

would then be a d u p l i c a t i o n i t s e l f .

The d i ip l icat ions generate no eleraents in ?, S^ that

are not already i n S, . Hence, the dupl i ca t ions add no

k2,

d l s t i n c t eXemimts t o S , S . Since t he re are eXements i n

S, S^ irtilch do not appear i n S, i t follows t h a t they must be

accounted for entireXy by the non-dt;?)lÍcated elements i n

S^ * Hence, we oouXd prodtice the sarae group S, S^ by con-

s t r u c t i n g the product group of S. and the non-duplicated

eXements of S^ with the i d e n t i t y element. The element s,

i s one of the non*dupXicated eXements as i t was so chosen.

I t foXXows t h a t the se t of non-d t^ l i ca t ed eXements and the

i d e n t i t y form a subgrotap of G. Therefore , our conclusion

must be t h a t s^ i s an element of a proper subgroup of S^ ,

which i s absurd . Since our conclusion was obtained by a s -

stimlng dtipXÍcations, then our assuisption must be wrong and

t h e r e can be no d u p l i c a t i o n s . The product group of S. and

S^ i s then S^^ S . X S^ , the d i r ec t product of independent

cyc l i c groups and of order p . I f S - S, X S = G

then the proof i s complete. I f i t i s no t , then an elemtent

s^ of per iod p ^ , where n 4 m ^ i m^<m can be chos en and

the same argument app l ied . Eventual ly , G can be expressed

as the d i r e c t product of ndependent cyc l i c groups,

G = S, X S^ X.e X S^ , where K (i ^ l , 2 , . . . , n ) i s

generated by an eleraent of per iod p

k* Conclusion; An Abelian group G expressed as (1) i s said

t o be of type (m , , m^, . . . . . . . mj where m=ra,^-ra^ -ra,,

and m. t m^i i r a ^ > 0 .

Example; - The Abelian group of order eight, whose multipli-

cation table appears on page 21, is the direct product of

k3 the Ixidependent cycXic groups generated by £ and t,.

Theorem X5

•*•• Statement; I f two AbeXian grotaps of order p*^ have an

eqtiaX nttmber of eXements of the same pe r iod , then t h e two are

sÍiapXy isomorphic.

^* Condi t ions; I t wiXl be necessary to show tha t the two

condi t ions f o r siiapXe isomorphism are s a t i s f i e d . We sha l l

consider the th ree pos s ib l e cases ;

(a) when both of the groups a re c y c l i c ;

(b) when bo th of the groups are non-cyc l i c ;

(c) when one of the groups i s cyc l i c and the other i s

n o n - c y c l i c .

3^ Proof; The proof sha l l be divided in to th ree p a r t s .

(a) I f the two Abelian groups are c y c l l c , then they

are siiî^^ly isomorphic.

(b) I f both of the groups are non-cyc l i c , then each i s

the d i r e c t product of independent cyc l ic groups (Theorem II4.).

Also, the independent cycl ic groi^)s of one of these non-

c y c l i c groups w i l l match the independent cycl ic groups of

the o the r non-cycl ic group one-to-one v/ith respect to order

s ince the two non-cycl ic groups have an equal number of e l e -

ments of the sarae o rde r . In o ther words, the c: c l l c groups

w i l l f a l l i n to p a i r s . Then, by the sarae argument as in ( a ) ,

t he c y c l i c groups are isoraorphic, and s ince the non-cycl ic

groups a re the d i r e c t product of the independent cyc l ic

10».

grot^s (whlch have only the i d e n t i t y element in common), then

the non*-eyoXic AbeXian groups must be isomorphic.

(c) I f one of tho Abelian grotips i s c y c l i c and the

o ther non^cycXic, then the non-cycXic group can be expressed

as the d i r e c t product of independent cycXic groups. Now

s i n c e every subgrotjp of a glven group i s cycXic and s ince

the two AbeXian groups have the same number of elements of

the same p e r i o d , then the argument of (b) appl ies t o t h i s

s i t t i a t i on and the two Abelian groups must be simply isomor-

p h i c . Hence, a l l three cases have been considered, and the

proof of the theorem i s complete.

A Survey of A l l Poss ib l e Abstract Abelian Groups

IVe are now equipped to make a svirvey of a l l p o s s i b l e

abstract Abelian groups* The two theorems which make t h i s

survey p o s s i b l e are;

(1) Every Abelian group i s the d irect product of i t s

Sylow subgrot5>s; i n other words, subgroups of the form p*" ,

where p i s a prime number and m i s a p o s i t i v e i n t e g e r .

(2) Every Abel an group of order p i s the d irect

product of independent c y c l i c grot^s .

I t i s we l l t o note that the tv/o above mentioned

theorems are independent of each other and considered sepa-

r a t e l y produce important r e s u l t s ; when considered simultane-

o u s l y , however, they become a very powerful device for raak-

ing a conplete survey of abstract Abelian groups.

It5 ^ For ea ampXe, consider an~AbeXian group G of order {

g = 529*200 2 • 3 • 5^ 7^ , By (X) G is the direct produet

of SyXow atibgroups of order 2 *, f , 5', and 7''. By (2) we

can ejpress each of theso SyXow subgroups (each in a number

of ways which is oompXetely determined by its power) as the

direct product of independent cycXÍc groups. Conversely,

given any positive integer g , we could certainly construct

aeveral abstract Abelian grotaps of that order. It is now

evident that the problem of determlning all possible abstract

AbeXian grotaps reduces to the probXera of determining all

possibXe prime power grot gps.

It foXlows from (2) that every eXement £ of a prime

power (p* ) grot$) can be eispressed as

s = s, s^ .....s^ , where

G = {s." x{s»)X X X and 1 i a. 4 p*"*'

i f p*"' i s the per iod of s. . G i s of type (m, , m ^ , . . . . , m^).

Let us stippose, however, t h a t

(3) s*' s^' B; = S = *'" s''; s^'where b^ 5 a^..

Now if we rewrite this as

[s ' s^J .•.•.6^"- s -fs."' s''") s.''** and consider

the portions In brackets, we can certainly say that

s*" s ': s^ s / or the order of {s. ) X {s \

would not be p . Now f we rewrite (3) as

f(í'sr)sr] • . • . C = s= [(s ' s^)s^] . . . .

and consider the bracketed portions, we can say that

(s*' sf )s? / (s'' s '' ^s,' or the order of

S^

k(> (•. ) ^ ÍK)^ ~í«»3 ''ovLlå hot be p'"'--**^ This process

can be continued and eventually we can say, as a consequence

of this, that s = 8*' s*" .....s*** is a tanique representa*

tion of the eXement s. The eXements s,, s^, ...., s are

a aet of ind^endent generators of G. The question might

arlse as to whether there can be another set (with different

periods) of independent generators of G. This is obviousXy

not possibXe; we wouXd then be speaking of an AbeXian grot^)

of a different type and hence of a different AbeXian group.

^,^ Therefore, a set of independent generators of an

ÅbeXian grotq) exists If end only if the grotp can be ex-

pressed as the direct product of independent cyclic groups.

Now let us recall that every grotjp is isomorphic

nrith a permutation group of equal order. Hence, an abstract

AbeXian grot^) G (of order p**') exists for ever ^ possible

distinct eiîpression of m as a sum of positive integers. In

view of the previous discussion of independent generators

and reforence to theorem 15 (pago k3) it Is seen that there

exists only one abstract Abelian group for each type. We

can now state and prove the following theorera.

Theorem l6

^* Statement; The number of abstract Abelian groups G of

order g = p ^ Px" •••••P^ > where p, , P, , , P, aro

distinct primes and a., a^, , a^ are positive integers,

Is equal to (number of part tions of a, into positive

_ k7 aummands) (ntomber of part i t ions of~ã^ into pos i t ive sum-

mands).. , . , . ,(nt2mbep of part i t ions of a into pos i t ive

atiamands)«

^* ££22£s Every such group G i s the direct product of i t s

SyXow subgrot^s of orders p " , p **" , , p* . Any of

these SyXow stibgrotgps,'^ for instance p * * (i = X, 2 , . . , . , s ) ,

may be of any type (a;, , a . , , a,^ ) i f a . = a,, +

^u -^ *•• ^i • However, one must consider a- = a«: to be

a par t i t i on of & into pos i t ive suraraands. By our taking

evei^ possibXe part i t ion of each £c $ every poss ible set of

Sylow subgroups of order g i s obtained. The direct products

of these sets give every poss ible Abelian group G of the

order g.

Bxample 1; The ntmiber of abstract Abelian groups of order

p* p' PI i s 2-3'3 = 18, In th is problem a ,= 2 , a^= 3 ,

a, = 3 .

It is well to note that the following discussion

refers to the powers of the primes and not the súbscripts

as there is no relation among them.

2 =2-^0 =1-*- 1

3 = 3+0 = 1 - ^ 1 + 1 = 2 +1.

It is obvious that there are two partitions of two

into positive summands and three partitions of three into

positive summands. Thls does not iraply that there are four

partitions of four into positive suramands, etc, as the next

IfS

•xa ^Xe wiXX iXXus"trãte .

Then, by t h e theorem, t h e r e must be 2-3-3 =X8

abatract AbeXian groups of o r d e r p* p ' p* , I t i s evident

from t h l s exampie t h a t t h e number of a b s t r a c t AbeXian g ro t^ s

of any p a r t i c u X a r o r d e r depends ent i reXy on t he powers . The

pr ime numbers a r e a r b i t r a r y ,

We have t a k e n every possibXe p a r t i t i o n of t he power

of each p . Now Xet a, = 2 , a ' = 1 + 1 , a^ = 3 , a; = 1 + 1 + 1

a ; - H - 2 , a^= 3 , ^3 1 + 1 - ^ 1 , a," = 1 + 2 . I f we t a k e a l l

p o s s i b l e d i r e c t p r o d u c t s of t h e SyXow subgroups , we should

have 1 8 .

0, =p , p /

0,-. p

o.

A.

G. = p, p.

p G = P ' I

< ' p. '

P P 2 3

3

G = p • p** p*'

G = p"' p^

G - p*" p * 5" . 2

G = p*' p*^

G = p'"' p**' a. 1

A. z

I

G- = p"' p

0 = p*' p" ' 7 1 Z

P^'

^5

G = 'S

G =

G = fS

G =

G ^ ' 7

G =. '8

p"'

I

«

P I

P< I

P"'

2

2

p^; a

P '

p«. 3

E x a i ^ l e 2 ; Determine t h e number of a b s t r a c t Abol ian groups

of o r d e r ^, = p p • I n t h i s oroblem a \ and a 6 .

I j . ^ i|. + 0 - 3+-1 = 2 + 2 - 2 1 - 1 - ^ 1 = 1 ^ - 1 + 1 - ^ 1

6 = 6 + 0 - 5 - ^ 1 - l i - ^ 2 - I}.-M+ 1 - 3 -«-3 ^ 3+2 + 1

= 3 - * - l + - l ^ l - 2 + 2 + 2 - 2 + 2 + 1 + 1 - ^ - ^ - H - l ^ l - H Í i .

= 1 + 1 + 1 ^ - 1 + 1 + 1 .

1 9

There a r e f i ve p a r t i t i o n s of four in to p o s i t i v e suraraands

and e leven p a r t i t i o n s of s ix i n t o p o s l t i v e sumraands, lîence,

t h e r e a re 5*11 = 55 a b s t r a c t Abelian groups of order

P. P .

Theorem 17

l^ Statement; The number of d i s t i n c t types of abs t r ac t

Abelian groups of order p*^, where p i a a prime number and

m a p o s i t i v e i n t e g e r , i s equal to the number of p a r t i t i o n s

of m i n t o p o s i t i v e summands.

^* Oondi t ions; Let G be an Abelian grot^ of order p* and

type (m, , m^, m^, , m, ) where m = m,^m^+ . . . . + m^ .

3 . Proof; I t follows from the theorera t h a t G i s the d i r ec t

product of cyc l i c subgroups of orders p***', p"^*, , p'*'*'.

fíence, t h e r e s an a b s t r a c t Abelian group corresponding to

the type (ra,, ra^, m , . . . . . . m^).

Eut m can be expressed as p o s i t i v e summands in

severa l d i f f e r en t ways; i t follows tha t for each vay of ex-

p re s s ing m as a p o s i t i v e summand there corresponds an ab-

s t r a c t Abelian group of t ha t t ype .

Exaraple; Let us consider the abs t r ac t Abelian groups of

order p where m - $.

5 = (1 + 1 + 1-I-1-»-1) so t he re s an abs t r ac t Abelian

group of order p and type ( 1 , 1 , 1, 1, 1 ) . Also,

5 = ( 2 + 1 + 1 + 1 ) so there i s an abs t rac t Abelian rroup

of order p and type (2, 1, 1 , 1 ) .

.S T l A n N - : ' v! UJLU^UÍi U t t U ^ K l

50

5 » {3 + X +X) 8o there i a an abs^tíãct IbeXian groi?) of

order p^ and type ( 3 , i , X).

5 = ( 2 + 2 + X) so the re i s an a b s t r a c t AbeXian group of

orú^rp and type (2, 2 , X).

This proceas can be continued for every p a r t l t i o n

of m i n t o p o a i t i v e summands.

The Ntimber of Blements of a Given Period

i n an Abelian Grot;^ G

Let us f i r s t consider the ntaaber of eleraents of

pe r iod p i n an Abelian group G of order p*" and type

f**,* ^z$ • • • • • ^ ) • ^®t £ be an element of per iod p in G,

or

s = s , s^ . . . . . . s ^ , where s , ,

S2.#*»#.» s form a set of independent generators of G and

a- < p" * (i == 1, 2, ...... k). Since £ is of order p, then

s = 3/ s * .....s **" = , Hence, we

can say that a p . is a multiple of p*"* or that a is divis-

ibXe by p * . Hence p * < a- < p*"* . Ke must now determine

how many elements of G sat sfy the above condition or hov/

many ways we can solve the following equation; »n.-1

a = Ap * such that the right side is

never greater than p***' . It is clear that A can then have

p values, namely A=l, 2. .....p. Substitution into the

original equation yields a total of p satisfactory ele-

ments (Including the identity). Hence, there are p * -1

5i eXements of per iod p in an Abelian /j-roup of order p* and

type (ra, , m^, , m j .

We s h a l l take a slightXy d i f fe ren t approach in d e t e r -

ffiinlng the number of elements of any period p*" i n an

Abelian group of order p*" and type (m, , m^, , m J . The

number of elements of per iod p'*' i s cerfcainly the difference

between the order of the subgroup cons i s t lng of a l l elements

each having a per iod which i s a f ac to r of p*' and the order

of the subgroup cons i s t i ng of a l l elements each having a

per iod which i s a f ac to r of p**" . Now we sha l l determine

the o rder of these two subgroups. The order of the f l r s t i s

p , where b ^m^ + m . ^ + . . . . . m ^ i- m^_^, + a se t of the remain-

ing elements . In t h i s equation m. I s the f i r s t nuraber in

(m, , m ^ , . . . . , , m^) l e s s than or equal t o a and ra,^^,= 0 . In

o ther words, the elements of a l l of the cyc l ic groups of

o rder l ess than p* are included; a l so included are sorae of

the eleraents of the cyc l i c groups of order p , where

m > m. , This se t of elements l a s t raentioned obviously con-

s i s t s of a l l the eleraents v/ith per iods l e s s than or equal

t o p . Now (1) can be wr i t t en as b =ra- -i- m. + ra^ i-ra ^^ ••- a

( i - 1 ) .

The order of the subgroup conta ning a l l e lenents

with per iods t h a t are f ac to r s of p* can be determined by

the same argtmient.

Exaraple; Let G be an Abelian ^xovp of order 3 and t : p e

52 ! ( IS , 10 , 8, 5 , H). Plnd^th. nuab.r of elenwnts of period

The stibgrot:^ G ' cons i s t ing of a l l elements with

periods that a r e f ao to r s of 3 i s of order 3**, where

b = 8 + 5 + 2 + 0+ 9(2) - 33 8o the order of

G i 8 3 . The order of t h e subgrotap G" cons i s t ing a l l

eXeaienta with with per iods t h a t a re f a c t o r s of 3* i s 3** ,

whi»pe

b ' ^ 8 + 5^ -2^0^-8 (2 ) - 31 so the order of

G i s 3 . Henoe, t he re are 3 - 3 eXements of period

fp, 3 i i i G^

Prope r t i e s of an Abelian Group G of Order

p and Type ( 1 , 1 , . . . . . , 1)

Since t h i s i s a group of type ( 1 , 1 , . . . . . . 1 ) , a l l

of the elements of G (excluding the i d e n t i t y element) are of

pe r iod p . Hence a subgroup of G of order p (1 < a < m) i s

a l s o of type ( 1 , 1 , « . . . . , 1 ) . Since a l l elements of G

(excluding the i d e n t i t y element) a re of per od p , t he re are

p - 1 elements of G of per iod p . Hence G i s the d i r e c t

product of c y c l i c subgrotip each of order p , and i t follows

t h a t every element of G may be included in a set or se t s of

genera tors of G.

P i r s t , we s h a l l consider the number of ways in which

a se t of generators of G may be chosen. Let s, , s^, , s ^

represen t t he elements of the group G which are of period p .

53

Since G contains p"*" -1 elements of order p, the first gen-

erator, s, , may be chosen in p"-l ways. Thenís,) contains

p-X elements of period p. fíence there are p'^-p elements of

period p remaining in G; thus the second generator s of G

may be chosen in p"-p ways. The group {s,, s > contains 2

p -1 elements of order p; there are, then, p'^-p^ elements

of period p remaining in G. The third generator s, may be

chosen in p'-p ways, and the group {s, , s^, s^} contains

p -1 elements of perlod p; so there are p*"-p' elements of

period p remaining in G. This process continues until all

the elements of G are contained in {s , s^, , s ^ .

Hence, the set s^, s^,....., s j of generators of G may be

chosen in (p^^-l) (p'^-p) (p* -p'') (p'"-p" ' ) ways.

As an example , le t an Abelian group G of order p'^^

3 - 2 7 and type ( 1 , 1 , 1) be considered. Let l and the 26

l e t t e r s of the alphabet designate t h i s group. The l e t t e r s

of the alphabet a l l represent eleraents of period p . The

elements of the grotp are 1, á, K, îj, H, ^, î*, ^, k , t , j - ,

k, i , m, n , e-, p-, q, r-, -s-, t-, Ur, v , w, at, y , -&. The f i r s t

generator s, can be chosen in 26 ways. Let us a r b i t r a r l l y

choose s, ^ £ and M = 3., a, b . llaric these eleraents out

( / ) . Froffl the reraaining 2i| elements l e t us a r b i t r a r i l y

choose Sj - c and (s^) = 1 , c , d, remerabering tha t (s. and

{s^) are independent . The eleraents of {s." X {s^^ w i l l

be nine in nuraber (includina^ the i d e n t i t y ) and ^Adll be

5k âetermined by the elements of {s,) and {s , Thê

nine elements are 3L, a, b, c, d, ac, ad, bc, bd. Mark the

Xaat six out (\). The last four (certainly distinct from

the firat five) can then be designated as ac = e, ad f,

bc - g, bd = h. Kow 83 can be arbitrarily chosen from the

•aet i, 3, , y, z. Let us choose s, = i and (s^} = 1,

i, J. Then ^s.) X {s^^ X (s,] consists of 27 distinct

eXements, nameXy ttie 27 eXements of G. The elements not

aXready mentioned are a = k, aj = 1, bi = m, bj n,

ci = o, cj = p, di = q, dj - r, ei - s, ej = t, fi - u,

^i = V, gi ^ w, gj = X, hi = y, hj z. Mark these out (-).

Hence, all possibilities have been considered, and

we have (26) Í2Í^) (18) = (p'-l) (p^-p) (p*-p* ) = 11,232 ways of

choosing an ordered set of generators of G. A given set of

m generators of G can be arranged In m! different orders.

Hence, the number of distinct sets of generators of G is

mî

In the example, the number of distinct sets of generators of

G is = ^^^ljf^^^'^^) = (26)(Í|)(18) =1872.

Theorem 19

1. Statement; The ntmiber of subgroups of order p ' ^^d l a^m) (p'"- l ) (p*"-p) (^"-P*" )

of G (of type 1, l , . l , ) i s equal to ^^-.i^) (^-.^ ') ; _ _ (p-.p—)

2« Condi t ions; A subgroup of order p*^ of G i s of type

( 1 , 1 , . . . . . . 1 ) . Tiiis i s t r u e because i t has already been

55

shown that all the eleraents of G (excluding the identity)

are of period p. Hence, any subgroup of G must be the direct

product of cyclic subgrotips of order p, and It follows im-

mediately that the subgroup is then of type (1, 1, , 1).

3» Proof 8 A subgroup of G of order p'*' has £ gemrators

of period p. The first generator may be chosen from G in

p -X ways; the second in p' -p ways; the third in p'-p^ways;

; the a~ in p -p ways. Rence, a set of £ genera-

tors of a subgroup of order p*" of G may be chosen in

(p*"-!) (p*^-p)...,. (p'*'-p' " ) ways, These sets are called

ordered sets of generators of the subgroup. An ordered set

of generators of a given subgroup of order p *" may be chosen

in (p -1) (p' -p) (p'*'-p* ). . . . . (p '-p*') ways. Hence, the above

product represents the total number of subgroups of order

p *• (not necessarily distinct).

Hence, it is now obvious that the number of distinct

subgrotps of order p*^ of G is equal to the quotient

(p--l) (p* -p) (P^-P' ) (P"-P*-* ).

Exanple ; Let G be an Abelian group of o rde r g = 3 = 2 7 and

t ype ( 1 , 1 , . . . . . , 1 ) . Let G b e a subgroup of G of o rde r

g = 3* -^9 and type ( 1 , 1 , , 1 ) . G ^ 1, a , b , c , d, e ,

f, g , ^h i * U ^s 1* i, n , o , p , q, r , s , t , u , v , w, x , y ,

z , Vie s h a l l now proceed t o choose c e n e r a t o r s of subgroups

of o r d e r 9 frora G. The f i r s t g e n e r a t o r obvious l : raay be

ehosen i n p " - l " 27-1 - 26 v/ays. Let us choose i t as a_.

56

Then {a> wiXl contain 3 elements, say 1, a, b . Then the

second genera tor may be chosen in p'^'-p = 27-3 = 2l\. ways.

Let us choose i t as £ . Then {a , ^ - 1, a, b , c, d, e, f,

g, h , which i s a group of order 9» Hence a set of generators

of p may be chosen in (26) (2i^) =62lj. ways. However, these

s e t s a re not neces sa r i l y d s t i n c t . Consider a p a r t l c u l a r

subgiN>up of G of order p = 9» Let G' - t h i s group of order

p = 3 = 9 . G ' 1 , a, b , c , d, e , f, g, h . Prcm t h i s

group of 9 elements we raay choose two generators of G

(remembering t h a t a l l except the i d e n t i t y are of period 8)

In (p**"-!) (p^'-p) ways, Let us choose the f i r s t generator as

a . Then (a) = 3., a, b . This generator could have been

chosen in p**'-l = 8 ways. Now the second generator raay be

chosen i n p*^-p = 9-3 = 6 ways. Let us choose the second gen-

e r a t o r as £ , Then { a , cS ~ 1, a, b , c , d, e, f, g, h .

Hence, we can choose two generators of G from G in (8)(l6) =

lí.8 ways, A l i t t l e thought w i l l raake I t evident tha t these

hfi ways a re included In the 62l| ways of choosing two gen-

e r a t o r s for G. îlence, the nuraber of d i s t i n c t subgroups o^ 62li

(of G) of order p i s -Trg = I 3 .

Corollary

1 . Statement; The number of subgroups of G of order p i s

equal to the ntimber of subgroups of order p

2 . Proof 5 I f t h i s i s t r u e then

( P " - 1 ) ( D ' " - P ) ( P " - P M . . ( I > - - P " ) J P " - 1 ) ( P " - P ) ( P " - P - ) . . ( P " - P " " ' )

p - l ) ( p -p)(p -p ) . . ( p -p ) (p - l ) ( p -p)^p -p ) . . ( p -p '

(p ' '^- l )p(p '""- l ) . . .p '*"(p '""- l ) _ ( p ^ - l ) p ( p ' " " - l ) . . . . p " " " ( p " " - l )

(p'^-l)p(p'^--1)....p'*-' (p-1) (p'-^.l)p(p'^'^''-1)...p'"-'^''(p-1)

(p . l ) (p--^. l ) (p- ' . l ) (p--^. l ) (p--^' . l ) . . (p '^-^. l ){p^"-l ) (p^- l )=-

(p-1) (p'^'^-1) (p" -' -1) ( p ' - l ) (p' *' - 1 ) . . . (p^'^-1) (p'"" -1) (p^-1).

Conslder the second equat ion . I t i s obvious tha t the cross

products must be i d e n t i c a l . We make the following observa-

t i o n s :

(1) Both cross productx have identical greatest and

least factors, namely, p -l and p-1, respectively.

(2) p'"'"'-l is the predecessor of p'"''*-l, and p *-1 is

the predeces3or of p -1.

(3) The factors of each cross product are consecutive

"powers of p" factors.

Hence, the cross products must be identical and the

proof is complete.

Exaraple: Let m - 6, k - 2, and m -k =4.

By the corollary thø number of subgroups of G of order p'

is equal th the number of order p .

(t>--i)(D -p) \r^-^}ir--rVv^-r>^nv^-v)

(p^-i)(p^-p) (F'-I)(P''-P)(P''-P'"P*-P'^

(r)'--l)p(p l) ^ (p*-nt>ÍD^-l)pMp''-lii^lL:J^

(p"-ljp(p-l) (p' -I)p(p'-1)P"(F'-I)p'íi-Í)

( p - l ) ( p ^ - l ) ( p ' - l ) ( p ^ - l ) ( t ^ - l ) ( p ' - i )

'- ( p - 1 ) ( p ' - l ) ( p ' - l ) ( p ' - l ) ( p " - l ) ( p ' - l )

BIBLIOGRAPHY

B u m s l d e , W» The Theory of Groups. cambridgeî University P r e s s , I 9 H 7

earmichael t Robert Danie l . In t roduct ion to the Theory of Groups of F i n i t e Order^ Boston. NewTorg? Ôlnn an Company, 1937#

R i l t o n , Harold . An n t roduct ion to the Theory of Groups of F i n i t e Order» Oxfordj tflaiversitíy P re s s , T^OtJ,

ÍJohnson, Richard E. F i r s t Course i n Abstract Algebra» New Yorkî Prent ice^Mal l , I n c , l9F3«

I»ederiBann, Walter , In t rodue t ion t o the Theory of P in i t e Group3» Edinburghî o l i v e r and Boyd,' l9'l4.9. New York; í n t e r s c i e n c e Pub l i she r s , 19if9»

Mathewson, Louis Clark. Elementary Theory of F in i t e Groups. Boston, New York; Houghton MÍfflin Company, 1^30•

M i l l e r , G» A.j B l i c h f e l d t , H. F . Î Dickson, L. E. ; F i n i t e Groups. New York: John Wiley and Sons. T n c , 19l6 , London* Chapman and Ha l l , Limited, 19I0.

Prasad, Gemiesh, Some Great Mathematiclans of the Nineteenth Century; Their Lives and Their Works in hree Volumes. Benares City ( í n d i a ) : Mahamandal Press , 193^4-

Smith, David Eugene. A Source Book in Mathematics. New York and London; "lîcGraw-fílll, 1^2 ";;

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