A History of Lagrange's Theorem on Groups
Author(s): Richard L. Roth
Source: Mathematics Magazine, Vol. 74, No. 2 (Apr., 2001), pp. 99-108
Published by: Mathematical Association of America
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VOL. 74, NO. 2, APRIL 2001
99
A History
ofLagrange's
Theoremon Groups
RICHARD
L. ROTH
University
ofColorado
Boulder,CO 80309-0395
Introduction
In group theory,the resultknownas Lagrange's Theoremstatesthatfor a finite
groupG theorderof anysubgroupdividestheorderof G. However,grouptheoryhad
notyetbeen inventedwhenLagrangefirstgave his resultand thetheoremtookquitea
form.Lagrange'sTheoremfirstappearedin 1770-71 in connectionwiththe
different
problemof solvingthe generalpolynomialof degree5 or higher,and its relationto
functions.
It was also anticipatedby some resultsin numbertheorya few
symmetric
yearsearlier.
In thisarticle,we explainthehistoricalsettingofLagrange'sapproach,and follow
thistrainofthought
intothetwentieth
Some generalreferences
century.
on thehistory
of grouptheoryare Israel Kleiner'sarticle"The Evolutionof GroupTheory:A Brief
Survey"in thisMAGAZINE[22] and H. Wussing'sbook, The GenesisoftheAbstract
GroupConcept[31].
Discussion
Preliminary
Let us firstreviewLagrange'sTheoremand its proof,as well as some otherresults
relevantto our discussion.Recall thatthe orderof a finitegroupis the numberof
elementsin thegroup.
THEOREMA (Lagrange'sTheorem):Let G be a groupofordern and H a subgroup
ofG oforderm. Thenm is a divisorofn.
SketchofProof. Suppose we list the elementsof G in a rectangulararrayas fol., am. If
b is some elementof G not in H thenlet the second row consistof the elements
bal, ba2, . .. , bam.If thereis an elementc notin thefirsttworowsthenthenextrow
will be ca1, ca2, . .. cam. This is continueduntiltheelementsof G are exhausted.One
mustthencheckthattheelementsin anyroware distinctand thatno tworowshave an
elementin common.It followsthatn = kmwherek is thenumberof rows.
m
lows:Let thetoprowbe thelistof them elementsof H: a, = e, a2, ..
Early proofsof Lagrange's Theoremgenerallyinvolvedthis "rectangulararray"
Note thattherowsof therectangular
explicitlyor implicitly.
arrayare simplytheleft
cosetsof H in G. In fact,mostcurrenttextsuse thelanguageof cosetsto provethis
theorem.One mustshowthatthesetof leftcosets(or rightcosets)formsa partition
of
thegroupG andthateach cosethas thesame numberofelementsas H. Notethatas an
elementary
consequenceofLagrange'sTheoremwe havethatthenumberof cosetsof
H in G dividestheorderof G as well. This numberis called theindexand is denoted
[G: H].
Frequently
in algebratextbooks,
thelittletheoremofFermatis provedas a corollary.
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MATHEMATICS
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FERMAT'SLITTLE THEOREM:If p is a primeand b is relatively
primeto p, then
bP-1 1_ (mod p).
Proof. The nonzeroelementsof 7/p7 forma groupof orderp - 1 undermultiplication,called (Z/pZ)*. If thecongruenceclass b in (Z/pZ)* has orderm, thenit
generatesa cyclic subgroupof orderm. By Lagrange'sTheorem,m dividesp - 1;
U
thus(b)P-l = (bm)k = 1, wherep-I = mk,and thetheoremfollows.
Also relevantis TheoremB, givenbelow.As we will see, itmightbe just as appropriateto call thisLagrange'sTheorem.
THEOREMB: Let G be afinitegroupactingbypermutations
on afiniteset S. Then
thesize ofanyorbitis a divisoroftheorderof G.
Proof. Let b be an elementof some particularorbitand H be the subgroupof G
stabilizingb. If c is anotherelementin thisorbitthenforsome r in G, rb = c. If scH
then(to)b = r(ob) = rb = c (in ournotationitis theelemento-in thestabilizerofb
thatacts first)and itis easilyseen thatthecoset r H consistsof preciselytheelements
mappingb onto c. Thus the elementsof the orbitare in one-to-onecorrespondence
withtheleftcosetsof H, hencethesize of theorbitequals theindex[G : H], which,
U
by ourconsequenceof TheoremA above,mustdivideG.
versionofthetheorem
Lagrange's
In 1770-71, Lagrangepublisheda landmarkworkon the theoryof equations,"Reflexionssur la resolutionalgebriquedes equations"[23]. (Note: Katz gives a useful
discussionof thispaper in section14.2.6 of [21].) His concernwas thequestionof
findingan algebraicformulafortherootsof the general5thdegreepolynomialand
moregenerallyforthenthdegreepolynomialforn > 4. The quadraticformulahad,
of course,been knownfora verylong timeand thecubic and quarticequationshad
been solvedin thesixteenth
centuryby algebraistsof theItalianschool.However,for
polynomialsof degreegreaterthanfourthishad remainedan open problemfortwo
centuries.Lagrangeobservedthatthe solutionsforthe cubic and quarticequations
involvedsolvingsupplementary
"resolvent"polynomialsof lowerdegreewhosecoefofthecoefficients
oftheoriginalpolynomial.He found
ficientswererationalfunctions
thattherootsof theseauxiliaryequationswerein fact"functions"
of therootsof the
originalequationthattookon a smallnumberof values whentheoriginalrootswere
permutedin theformulasforthesefunctions.
For example,thequarticwas solvedusinga cubicresolventpolynomialwhoseroots
could be written
as xlx2+x3x4 xlx3+x2x4 and XlX4?x2x3 wherex1, X2, X3, X4 weretheroots
oftheoriginal
IfthefourrootsX1,
polynomial.
X2, X3, X4
in all 24 posarepermuted
sible ways, onlythesethreedifferent
"values" typicallyoccur.For convenience,we
will omitthe denominator
2 in what follows.We list below the resultof operating
on the functionx1x2+ X3X4 by the 24 different
of the fourvariables.
permutations
The stabilizerof the functionis a groupof 8 elementsand the firstrow shows the
seven othervalues thatare equal to the originalone. Underneatheach value is the
The secondrowshowsthesetofeightvaluesarisingfrom
corresponding
permutation.
a different
way of combiningthe fourroots,all equal to x1x3+ x2x4; similarlyfor
thethirdrow.Underneatheach value is thecorresponding
We notethat
permutation.
in the second and thirdlines we have permutedthepositionsof thevariablesin the
same manneras was done in thefirstline.This correspondsto thestepin theproofof
o-mustactbeforethepermutation
TheoremB above wherethestabilizingpermutation
t thatchangesthe object. For example,referring
to the firsttwo equal functionsin
X1X2 +
X3X4
id
X1X3 +
X3X4
(12)
X2X4
(23)
X1X4 +
X2X1 +
X2X3
(243)
X3X1 +
X2X4
(23)(12)
= (132)
X4X1 +
X2X3
(243)(12)
= (1432)
X3X4 +
X1X2
X1X2
(13)(24)
X2X4 +
X1X3
(23)(13)(24)
= (1243)
X2X3 +
X1X4
(243)(13)(24)
= (123)
+
X4X3
(34)
XlX3
+
X4X2
(23)(34)
= (234)
XlX4
+
X3X2
(243)(34)
= (24)
X4X3 +
XlX2
(1423)
X4X2 +
X1X3
(23)(1423)
= (143)
X3X2 +
XlX4
(243)(1423)
= (13)
X3X4 +
X
(1324
X2X4 +
X
(23)(13
= (12
X2X3 +
X
(243)(13
= (123
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MATHEMATICS
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rows 1 and 2, we haveo- = (1 2), r = (2 3) and ro = (2 3)(1 2) = (1 3 2). We thus
have a 3 by 8 rectangular
array.The 3 "values" multipliedby 8 gives24 = 4! = IS4I.
The 4thdegreepolynomialwas solvablebecause therewas a "function"of 4 variables whichtookon 3 "values"whenthe4 variablesarepermutedin all 24 = 4! ways.
That is, these3 "values" were therootsof a cubic polynomial(whichit was known
how to solve); theserootscould be used to modifythe original4th degreepolynomial so thatitwouldfactorintoquadraticpolynomials.Usingthetheoryof symmetric
functions,
Lagrangeprovedthatifa rationalfunctionof then rootsof a generalpolynomialof degreen takeson r "values" underthe actionof all n! permutations,
then
thefunction
will be a rootof a polynomialof degreer whose coefficients
are rational
of thecoefficients
of theoriginalequation.
functions
Thus Lagrangereasonedthatto solve a 5thdegreepolynomial,one shouldtryto
in 5 variablesthattakeson 3 (or 4) different
finda function
typical"values" whenthe
variablesarepermuted
in all 5! ways.This wouldlead to a cubic (or quartic)resolvent
thatmighthelp to solve the originalequation.A similarapproachmightapply for
solvingequationsof degreen forn greaterthan5.
ifsuchfunctions
Lagrangewas unableto determine
exist.But he did come up with,
in essence,thefollowingtheorem.
THEOREM C: THEOREM OF LAGRANGE: If a function f (xI,
...
,
xn) of n vari-
ables is acted on byall n! possiblepermutations
ofthevariablesand thesepermuted
functionstakeon onlyr distinctvalues,thenr is a divisorofn!.
In fact,Lagrange statedhis theoremin termsof the degreeof the corresponding
resolventequation.Also, we notethatifn = 5, then3 and 4 arebothdivisorsof n!, so
thetheoremof Lagrangedoesn'tanswerthepreviousquestionas to whetheror nota
cubic or quarticresolventexistsforthe5thdegreeequation.
Lagrange'sproofof TheoremC consistedessentiallyof discussingsome special
thathe gave forthe firstcase was
to note thatthe treatment
cases; it is interesting
partlywrong,althoughit did givethecorrectidea fora proof.He said: let us suppose
thata functionsatisfies f (x', x", x"', xiv...) = f(x", x"', xx'
.. .). Such a function
(
x X
X" ....),
satisfiesf (xiv',xlll,x , x,
because we have permuted
thefirst3 variablesin thesame way; hence,he said, all thevalues will matchup in
pairs and thepossible numberof distinctvalues will be reducedto n'. However,the
involvedis actuallya cycle of length3, so in factin thisexample the
permutation
numberof values wouldbe dividedby 3, and notby 2.
Lagrangethensaid thatif theoriginalfunctionremainsthesame under3 or 4 or a
thentheothervalues will also have thatproperty,
and
largernumberof permutations,
thetotalnumberof distinctvalueswill be "I or "', etc.
Thus Lagrange'soriginalTheoremC mightbe regardedas a special case of TheoremB, wherethegroupG is thesymmetric
groupSn, theset S is theset of functions
of then variablesand
(or formulas)involvingn variablesformedby all permutations
thegroupactionis thatwhicharisesfrompermuting
thevariablesin thesefunctions.
in 1771 Vandermondealso wrote a paper ([28]) on the theory
Coincidentally,
of equationsthattook an approachsimilarto Lagrange's. The alternating
function
takes
on
two
values
when
the
variables
are
In
(xi
xj)
exactly
permuted.
I<i<j<n
Vandermonde's
paperwe findthisfunctionforthecase whenn = 3. It is used today
abstractalgebrabooks to studyevenand odd permu(forarbitary
n) in contemporary
thatstabilizeit formsthe alternating
tations;the set of permutations
group.As can
be seen fromour paper,theuse of polynomialfunctionsis historically
verymucha
- xj) is actuallyequal to
functionHl<i<j<n(xi
partof grouptheory.The alternating
the Vandermondedeterminant,
and it is probablethatthe originof thename comes
fromthisreference,
althoughVandermonde
(who elsewheredid do important
workon
...
H
)
VOL. 74, NO. 2, APRIL2001
103
did notexpressit as a determinant.
For thecase ofn = 5, Vandermonde
determinants)
gave a different
exampleof a functiontakingon two values underpermutations,
and
of fivevariablestaking
expressedtheopinionthattheredoes notexistsucha function
on eitherthreeor fourvalues.
Somelaterdevelopments
relatedto Lagrange's
work
Severaldecades later,Paolo Ruffinimade further
progressin Lagrange'sapproachto
solvingpolynomialequations.His book of 1799 [25] includedan informalproof(by
Ruffinishowedthattheredoes notexist
example)of Lagrange'sTheoremC. Further,
of 5 variablestakingon threevalues or fourvalues.Thus the"converse"
anyfunction
to TheoremC is false.It seemsappropriate
to call this"Ruffini's
Theorem."In modern
terminology
thisshowsthattheconverseto Lagrange'sTheorem(TheoremA) is false
because the symmetric
groupon 5 lettersof order120 has no subgroupof order40
or 30.
Ruffinialso claimedto have proved(as a consequence)thatthe5thdegreeequation(and in generalthenthdegreeforn > 5) was notsolvable.His workdrewmuch
criticismand even thoughhe publishedseveralmoreversions,his proofis generally
regardedas incomplete.(A moresatisfactory
proofwouldbe givenlaterbyAbel in the
1820s.) A friendly
responsecame fromAbbatiin 1802 [1] who gave some suggestions
to improveRuffini'sproof.Abbati'snoteincludedan extensiveand thoughtful
proof
of Lagrange'sTheoremC. Accordingto HeinrichBurkhardt
[5] thiswas thefirsttime
a completeproofwas given.It resembledtheproofof TheoremA givenabove, and
presenteda rectangular
arrayof terms(as illustratedin our earlierdiscussionof the
function
X1X2
+ X3X4).
The givenfunction
The
was actedon bythen! permutations.
firstrow gave the list of functions(arisingfrompermutations)
thatare equal to the
originalfunction.The nextrow startedwitha value thatwas notin thefirstrow and
consistedof all subsequentpermutations
of thisfunction.Because themodemnotationused in theproofofTheoremB was notavailable,a lengthier
discussionwas used.
Abbatitooksomecare to explainthatwhatmattered
was thepositionsofthevariables
in thefunctionand theway theywerecombined.Then,as in theproofof TheoremA,
each row was shownto have thesame numberof elements,withno tworowshaving
anyelementin common.
Furtherdevelopmentswere obtainedwith Cauchy's important1815 paper [7],
whose titleroughlytranslatedintoEnglishreads "Memoiron thenumberof values
thata functioncan acquire (when one permutesin all possible ways the quantities
it contains)."This paperlaunchedpermutation
grouptheoryas an independent
topic
even thoughthenotionof a groupdid notappearin it. The paperwas notconcerned
directlywiththetheoryof equations.Cauchyincludeda proofof TheoremC similar
to Abbati's,pointingoutthatthepermutations
areto be appliedto the
fixinga function
thepositionsof thevariablesand nottheirindices.He wenton to generalizeRuffini's
oremas follows:If thenumberof values of a non-symmetric
functionof n quantities
is less thanthelargestprimedivisorp of n thenit mustbe 2. In the 1820s Abel cited
forthespecial case thatn = 5
and used thistheoremfromCauchy'spaperspecifically
in his workon theunsolvability
of thequintic(see [2, p. 31] of Oeuvres). Although
Ruffini
was notexplicitlymentionedin Abel's paper,Abel's proofof theunsolvability
of the5thdegreepolynomialthusreliedindirectly
on theworkof Ruffini.
Galois introduced
thetermgroupforpermutation
groupsin a paperon solutionsof
polynomialsby radicalsin 1831 [14, pp. 35-36]. He didn'texplicitlymentioneither
formof Lagrange's Theoremin any of his papers,but in the famousletterwritten
the nightbeforehis death [15], he did includethe suggestiveequationforthecoset
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MATHEMATICS
MAGAZINE
decomposition
G = H + HS + HS'+
.
(1)
These workswerenotwidelyknownuntiltheywerepublishedin Liouville's Journal
in 1846.
Theory
ofpermutation
groups
Almost 30 years afterCauchy's 1815 paper,Cauchy again took up the subject of
permutation
groups.His paper of 1844 [8] did not take the "values of a function"
approach;ratherit dealt directlywithpermutation
groups.Keepingthe older use of
the word "permutation"
to referto an orderedarrangement
of symbols,he used the
term"substitution"
to referto a permutation,
and "systemof conjugatesubstitutions"
(systemede substitutions
conjugees)to referto a permutation
group.This was defined
here and in laterworkson the subjectmerelyas a set of permutations
closed under
to definea group:compositionis associative,
composition.This is of coursesufficient
and since we are dealingwithfinitepermutations,
the existenceof the identityand
inverseoperationswill be necessarilyimplied.(It is perhapsunfortunate
thatin many
modembooksthedefinition
ofgrouponlyliststhreeaxioms,namelyassociativity
and
theexistenceof theidentityand inverseoperations.The closureproperty,
whichwas
reallytheoriginalproperty
used to definepermutation
groups,is hiddenin theterm
"operation"or "binaryoperation.")Cauchythenprovedthefollowingtheorem(translatedintoEnglish):"The orderof a systemof conjugatesubstitutions
on n variables
whichone can formwiththese
is alwaysa divisorof thenumberN of arrangements
variables"[8, p. 207]; i.e. theorderof a subgroupof thesymmetric
groupSn is a divisorof n!. Thus we nowhad TheoremA (Lagrange'sTheorem)forthecase thatG is
thesymmetric
group.
Cauchy'slongpaperof 1844was followedbya seriesofshorter
papersoverthenext
coupleofyearsthatfurther
developedthetheoryofpermutation
groups.Theyincluded
showingtheconnectionbetweenTheoremA and TheoremC. In [9], he showedthat
the set of permutations
fixinga functionformsa permutation
group(i.e. a subgroup
of Sn, whichtodaywe call thestabilizerof thatfunction)and that,conversely,
forany
such subgroupH of Sn thereis a function
whose stabilizeris preciselythatsubgroup.
Given a subgroupH of Sn, we can build a functionof n variableswhose stabilizer
is H in thefollowingway. Let s = alxl +
+ anXn, wherea,, .. ., an are distinct
numbers.Let sI, 2, . .. , st be the images underthe t elementsof the subgroupH
(notewe may assumes = sl); thenconsidertheproduct5I52 ... St. Withappropriate
whose stabilizer
assumptionson thecoefficients
a,, a2, ... , an, thiswill be a function
is preciselyH. Cauchyrequiredthesecoefficients
to be nonzeroelementswhose sum
is notzero and his proofis a bitunclear,butifwe requireinsteadthatthecoefficients
be a set of nonzerointegerswhose greatestcommondivisoris 1, thentheresultcan
be shownusingtheunique factorization
of polynomialringsover a unique
property
factorization
domain.
Anotherexampleof a functionwhose stabilizeris H, also givenby Cauchy,arises
bytakings = xIx2x3 ... xn, and defining
sI = s, S2, .. . ,St to be theimagesofs under
the subgroupH; thenthe sum si +
+ st is a functionwhose stabilizeris H. In a
laternote [10] he showedthatif a functiontakes on m distinctvalues and the subgroupfixingthefunctionhas orderM thenmM = n!. Because everysubgroupis the
stabilizerof some function,
thisis a kindof "hybridform"of Lagrange'sTheoremin
whichCauchycombinedTheoremC and TheoremA forthecase of G = S. In this
.
VOL. 74, NO. 2, APRIL 2001
105
notehe also laid thefoundationsfortheidea of a groupactingon a set (as in TheoremB above). In particular
he showedthata groupofpermutations
on then variables
xi,
. . .,
xn could also be regarded as acting by permutationson the set of functionsthat
arisefroma particular
function
of thosevariablesunderpermutations
of thevariables.
The nextstepin thedevelopmentof Lagrange'sTheoremwas to see thatTheorem
A holds for any finitepermutation
group G. This resultmay be foundin Camille
Jordan'sthesis,publishedin 1861 [19]. This is a ratherlengthyand technicalpaperon
permutation
groups.In theintroduction
he citedLagrange'sTheorem,in the"hybrid"
formjust mentioned,callingit a theoremdue to Lagrangeand creditinghis proofto
Cauchy.Forty-five
pages laterin the midstof a complicatedcountingargument,
he
mentionedthathe wouldneed a generalization
of theLagrange'sTheoremand proved
that(in modemlanguage)theorderofa subgroupofanypermutation
groupdividesthe
orderof thegroup.Lagrange'sTheoremthenappearedin thisformin the3rdedition
of Serret'simportant
algebratextCours d'Algeibresuperieurepublishedin 1866 [27]
(in a laterchapter,the theoryis applied to the topic of the numberof values of a
functionof n variables).And it is also foundin Jordan'sinfluential
1871 book Traite
des substitutions
et des equationsalgeibriques[20].
Lagrange'sTheoremC is essentiallyTheoremA forthecase whereG = Sn,as we
have seen above: each functionhas its stabilizingsubgroupof Sn and each subgroup
of Sn may be regardedas the stabilizerof an appropriate
function.The extensionof
TheoremA to thecase of G, an arbitrary
permutation
group,was moregeneraland
mightseemto haveno analoguein termsoffunctions.
However,itis interesting
to note
thatin Netto'sbook of 1882 [24] he did give a translation
intothefunctionapproach.
"LehrsatzVI" in chapter3 states:If qoand 1 aretwofunctions
of thesame n variables
and the permutations
leaving pofixedalso leave * unchanged,thenthe numberof
valuestakenon by (o is a multipleof thenumberof valuestakenon by 1.
ingrouptheory
Otherdirections
While the theoryof permutation
groupsplayed a major role in the developmentof
generalgrouptheory,therewere a numberof otherimportant
sourcesof grouptheoryincludinggeometryand numbertheory.One connectionto numbertheory,
as we
notedin theintroduction,
is Fermat'sLittleTheorem.Eulergave severalproofsofthis
theorem.Of interesthere is his paper whose titletranslatedintoEnglishis "Theoin 1758-59 andpublishedin
remson residuesobtainedbydivisionofpowers,"written
In
it
a
the
lines
in
1761 [11].
he gave proofalong
indicated thebeginningofthisarticle.
He provedLagrange'sTheoremin essentiallytheusual way (therectangular
array)for
thecase thatG is (Z/pZ)* (themultiplicative
groupof theintegersrelatively
primeto
p, modulop) and H is thecyclicsubgroupgeneratedby b. Thus one could arguethat
in some sense,Lagrange'sTheoremappeared10 yearsbeforeLagrange'swork.The
theoremof Fermatgeneralizesin two directions;if,insteadof (Z/pZ)*, we consider
(Z/nZ)*, theclasses of integersrelativelyprimeto n, then W _ 1 mod n (whereso
is theEulerfunction).This was shownby Eulerin a paperwritten1760-61, published
in 1763 [12]. The otherdirectionis to regard(Z/pZ)* as a finitefieldandto generalize
thisto Galois fieldsGF(pn), in whichcase we have xpn-l = 1, and thiswas done by
Galois in 1830 [13]. Again,in bothcases, theproofsimplicitly
involveda rectangular
arrayapproachto a special case ofLagrange'sTheorem.
The developmentof theabstractapproachto groupsis discussedin [22] and [31]:
abstractgroupsgeneralizedpermutation
groups,variousgroupsarisingin numbertheas we've seen in theprevious
ory(includingthosearisingfrommodulararithmetic,
paragraph)and geometricalgroups.The abstractapproachto groupscaughton in the
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MATHEMATICS
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1880s. Of courseLagrange'sTheorem(TheoremA) and its proofwere by theneasily adaptedto abstractgroups.It is hardto pinpointthefirstabstractversion;but,for
example,in a paper of Holder in 1889 [18] on Galois theory,an abstractdefinition
forfinitegroupswas given.Lagrange'sTheoremwas provedby merelydisplayingthe
familiarrectangular
array.The theoremwas notidentified
byLagrange'sname,butthe
rectangular
arrayis creditedto Cauchy's 1844 paper.
In 1895-96, Weber'sLehrbuchder Algebra [30] was publishedin two volumes.
This became the standardtextformodemalgebraforthenextfewdecades. Volume
1 (1895) was themoreelementary
of thetwo volumes.As forgrouptheoryit treated
onlypermutation
groups.It includedTheoremA forpermutation
groups.The theorem
was creditedto Cauchy.The proof,however,was doneusingtheterminology
ofcosets
("nebengruppe").It was shownthattwo cosetsare eitherequal or disjoint,and stated
thatthe size of any coset equals theorderof the subgroup.Followingtheproof,the
symbolicequation
P = Q + Q7T1+ Q7T2+
+ Q7rj-l
(2)
was displayed(where P is the groupand Q is the subgroup)to give more insight
intothe proof,and thisequationwas creditedto Galois. Note thatGalois's version
(equation(1)) did notspecifya finitesum,however.This mayhave been thefirsttime
thatthetheoremwas provedusingthelanguageof cosets.
Weber'sVolume2 (1896) coveredmoreadvancedmaterialthanvolume 1; it began withabstractgroupsand provedLagrange'sTheoremagain,thistimeforabstract
groups,in section2 of chapter1. The proofwas essentiallytherectangular
arrayapproach(withoutexplicitlygivingthearray).The theoremand itsproofwerethenused
as motivation
theconceptof coset ("as in thespecial case of permuforintroducing
tationgroups")and a coset decompositionequation,similarto equation(2) was displayed.
of thenumTheoremC did notappearin Weber'sbook and theonlyconsideration
was the case of symmetric
ber of values takenon by a functionunderpermutations
functionsand the alternating
function(discussedin volume 1). A proofof Fermat's
LittleTheoremappearedin volume 1 (provedwithoutgrouptheory);in chapter2 of
volume2 the generalizationof theFermatTheoremforthe integersmodulo n was
provedusinggrouptheoryand Lagrange'sTheorem.
Twentieth
century
developments
in thetwentieth
coset terminology
was used in theproofof LaIncreasingly
century,
fromtherectangular
grange'sTheorem.It is notso different
arrayapproach,sincethe
rowsofthearrayarein factthecosets.Butitis a different
style,andwhiletherectangulararrayis usuallycreditedto Cauchy,thecosetapproachseemsto havebeen inspired
byGalois's cosetdecompositionequation(1). We saw thatthesetwohistoricalthreads
in Weber'sbook.
werebroughttogether
Accordingto Wussing[31], the firstmonographdevotedto abstractgrouptheory
was thatofDeSeguier,appearingin 1904 [26]. He showedthatthedoublecosetsform
a partitionof a group.Then as a special case, he got the coset decompositionof a
groupG and thensimplymentionedthat[G, A][A, 1] = [G, 1]. Here [G, A] denotes
theindexof a subgroupA in G so [G, 1] is theorderof G.
Althougha numberof authorscreditedthe theoremto Lagrange,manydid not
mentionLagrange'sname and it was some timebeforeit became widelyknownas
"Lagrange'sTheorem."Van derWaerden'sModerneAlgebra[29] was one ofthemost
influential
textsin algebra.It firstappearedin 1930. The coset approachwas used in
VOL. 74, NO. 2, APRIL2001
107
provingLagrange'sTheorem.Lagrange'snamedid notappear;however,in a footnote
itwas mentionedthatthecosetdecompositionequation(1), whichis frequently
found
in theliterature,
is due to Galois. In the"secondrevisededition"of 1937,thefactthat
anytwo cosets have the same numberof elementswas explicitlystatedand proved,
thusfillinga gap in thefirstedition.In theEnglishtranslation
(of thesecondrevised
edition),appearingin 1949,thetranslator
added a footnotestatingthatthistheoremis
also knownas Lagrange'sTheorem.
We mightcomparetwootherbooks whichalso appearedin 1937. Albert'sModern
HigherAlgebra [3] used thelanguageof cosets to provethetheorem;therewas no
mentionofLagrangeor otherhistoricalreferences.
On theotherhand,in Carmichael's
Introduction
to the Theoryof Groupsof FiniteOrder [6], theproofwas givenusing
the rectangulararray.Carmichaelcalled the theorem"FirstFundamentalTheorem"
(his chapter2 containedfivefundamental
theorems)buttherewas a footnotestating
"This has sometimesbeen called thetheoremofLagrange".
In 1941, Birkhoff
and MacLane's A SurveyofModernAlgebrafirstappeared[4].
This book became a model forundergraduate
modem algebratextbooksand helped
to attachLagrange'sname firmly
to thetheorem.Section9 of chapterVI is entitled
"Lagrange'sTheorem."It startedwithtwo lemmasshowingthateach coset has the
same numberof elementsas the subgroupsand any two distinctcosets are disjoint.
This led up to "Theorem18 (Lagrange): The orderof a finitegroupG is a multiple
of theorderof everyone of itssubgroups."Therewere variouscorollariesincluding
Fermat'sLittleTheorem.
Some modembooks applythegeneraltheoryof equivalencerelationsin connectionwithcosetsand Lagrange'sTheorem(forexample,see Herstein'sAbstractAlgebra [17]). One definesa relationon thegroupG bylettingaRb ifab-1 H. It is proved
thatthisis an equivalencerelationand therightcosetsaretheequivalenceclasses. The
of thegroupG. One of theearliestexamplesof this
rightcosetsthusforma partition
approachis foundin a book of Hasse, published1926 (see [16]).
We wishtothankthereferees
formanyhelpfulsuggestions.
Acknowledgment.
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Figure1 Problem5 fromUSA Mathematical
Olympiad(p. 167)