A STUDY OP ABELXAN GROUPS A THESIS IN MATHEMATICS by …
Transcript of A STUDY OP ABELXAN GROUPS A THESIS IN MATHEMATICS by …
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 )
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