Zero-sum problems in abelian groups

33 Pages·2006·0.37 MB·English

Expo.Math.24(2006)337–369
www.elsevier.de/exmath
Zero-sum problems in finite abelian groups: A survey
Weidong Gaoa,Alfred Geroldingerb,∗
aCenterforCombinatorics,NankaiUniversity,Tianjin300071,PRChina
bInstitutfürMathematikundWissenschaftlichesRechnen,Karl-FranzensUniversität,
Heinrichstrasse36,8010Graz,Austria
Received19December2005
Abstract
Wegiveanoverviewofzero-sumtheoryinfiniteabeliangroups,asubfieldofadditivegrouptheory
andcombinatorialnumbertheory.Indoingsoweconcentrateonthealgebraicpartofthetheoryand
onthedevelopmentsincetheappearanceofthesurveyarticlebyY.Caroin1996.
(cid:1)2006ElsevierGmbH.Allrightsreserved.
MSC2000:11B50;11P70;11B75
Keywords:Zero-sumsequences;Finiteabeliangroups
1. Introduction
LetGbeanadditivefiniteabeliangroup.Incombinatorialnumbertheoryafinitesequence
S = (g ,...,g) = g · ... · g of elements of G, where the repetition of elements is
1 l 1 l
allowedandtheirorderisdisregarded,issimplycalledasequenceoverG,andS iscalled
a zero-sum sequence if g + ··· + g = 0. A typical direct zero-sum problem studies
1 l
conditions which ensure that given sequences have non-empty zero-sum subsequences
withprescribedproperties.Theassociatedinversezero-sumproblemstudiesthestructure
ofextremalsequenceswhichhavenosuchzero-sumsubsequences.
∗
Correspondingauthor.
E-mailaddress:[email protected](A.Geroldinger).
0723-0869/$-seefrontmatter(cid:1)2006ElsevierGmbH.Allrightsreserved.
doi:10.1016/j.exmath.2006.07.002
338 W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369
TheseinvestigationswereinitiatedbyaresultofP.Erdo˝s,A.GinzburgandA.Ziv,who
provedthat2n−1isthesmallestintegerl ∈ NsuchthateverysequenceS overacyclic
group of order n has a zero-sum subsequence of length n (see [47]). Some years later,
P.C.Baayen,P.Erdo˝sandH.Davenport(see[138,45,143])posedtheproblemtodetermine
thesmallestintegerl ∈NsuchthateverysequenceSoverGoflength|S|(cid:1)lhasazero-sum
subsequence.InsubsequentliteraturethatintegerlhasbeencalledtheDavenportconstant
ofG.ItisdenotedbyD(G),anditsprecisevalue–intermsofthegroupinvariantsofG–
isstillunknowningeneral.
These problems were the starting points for much research, as it turned out that ques-
tions of this type occur naturally in various branches of combinatorics, number theory
and geometry. Conversely, zero-sum problems have greatly influenced the development
of various subfields of these areas (among others, zero-sum Ramsey theory was initiated
by the works of A. Bialostocki and P. Dierker). So there are intrinsic connections with
graph theory, Ramsey theory and geometry (see [119,4,12,13] for some classical papers
and [11,10,105,14,108,40,123] for some recent papers). The following observation goes
backtoH.Davenport:IfRistheringofintegersofsomealgebraicnumberfieldwithideal
classgroup(isomorphicto)G,thenD(G)isthemaximalnumberofprimeideals(counted
with or without multiplicity) which occur in the prime ideal decomposition of aR for ir-
reducibleelementsa ∈ R.Indeed,inthetheoryofnon-uniquefactorizationsithasturned
outthatthemonoidofallzero-sumsequencesoverGcloselyreflectsthearithmeticofa
KrullmonoidwhichhasclassgroupGandeveryclasscontainsaprime(see[96,Corollary
3.4.12]).Ontheotherhand,itwasfactorizationtheorywhichpromotedtheinvestigationof
inversezero-sumproblems,whichappearnaturallyinthatarea.Apartfromallthat,zero-
sumproblemsoccurinvarioustypesofnumbertheoreticaltopics(asCarmichealnumbers
[1],Artin’sconjectureonadditiveforms[19]orpermutationmatrices[135]).
Zero-sum problems are tackled with a huge variety of methods. First of all we men-
tion methods from additive group theory including all types of addition theorems (see
[96,136,137,142,112,107,109,103,133,124,9]). Furthermore, group algebras [74], results
fromthecoveringarea[164,132,80],fromlinearalgebra[32,31]andpolynomialmethods
[2,3]playcrucialroles.Moreover,inthemeantimezero-sumtheoryhasalreadydeveloped
itsownmethodsandawealthofresultswhichpromoteitsfurtherdevelopment.
The first survey article on zero-sum theory, written byY. Caro, appeared 10 years ago
in 1996 (see [23,24]). The aim of the present article is to sketch the development in the
lastdecadeandtogiveanoverviewoverthepresentstateoftheareaunderthefollowing
two restrictions. First, we do not outline the relationships to other areas, as graph the-
ory, Ramsey theory or the theory of non-unique factorizations, but we restrict to what
is sometimes called the algebraic part of zero-sum theory. Second, although since the
1960s zero-sum problems were studied also in the setting of non-abelian groups (see
[36,146,150,147,148,170,63,39,172]), we restrict to the case of abelian groups. Since
Y. Caro’s article has an extended bibliography on the literature until 1994, we also re-
fer to his bibliography and concentrate ourselves on papers having appeared since that
time.InSection2,wefixournotationsandterminology,andwegivethedefinitionsofthe
keyinvariants.Theninthesubsequentsectionswepresentthestateofknowledgeonthese
invariantsandontheassociatedinverseproblems.
Throughoutthisarticle,letGbeanadditivefiniteabeliangroupandletG•=G\{0}.
W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369 339
2. Preliminaries
Let N denote the set of positive integers, P ⊂ N the set of all prime numbers and let
N =N∪{0}.Forintegersa,b∈Zweset[a,b]={x ∈Z|a(cid:2)x(cid:2)b},andforc∈Nlet
0
N(cid:1)c=N\[1,c−1].Forarealnumberx,wedenoteby(cid:5)x(cid:6)thelargestintegerthatisless
thanorequaltox,andby(cid:7)x(cid:8)thesmallestintegerthatisgreaterthanorequaltox.
Throughout, all abelian groups will be written additively. For n ∈ N, let C denote a
n
cyclicgroupwithnelements,andletnG={ng |g ∈G}.BytheFundamentalTheoremof
FiniteAbelianGroupswehave
G(cid:1)C (cid:2)···(cid:2)C (cid:1)C (cid:2)···(cid:2)C ,
n1 nr q1 qs
where r = r(G) ∈ N is the rank of G, s = r∗(G) ∈ N is the total rank of G,
0 0
n ,...,n ∈ N are integers with 1<n | ... | n and q ,...,q are prime powers.
1 r 1 r 1 s
Moreover,n ,...,n ,q ,...,q areuniquelydeterminedbyG,andweset
1 r 1 s
(cid:1)r (cid:1)s q −1
d∗(G)= (n −1) and k∗(G)= i .
i
q
i=1 i=1 i
Clearly,n =exp(G)istheexponentofG,andif|G|=1,thenr(G)=d∗(G)=k∗(G)=0
r
andexp(G)=1.
Lets ∈ N.Ans-tuple(e ,...,e )ofelementsofGissaidtobeindependentife (cid:9)= 0
1 s i
foralli ∈[1,s]and,foreverys-tuple(m ,...,m )∈Zs,
1 s
(cid:1)s
m e =0 implies m e =···=m e =0.
i i 1 1 s s
i=1
An s-tuple (e ,...,e ) of elements of G is called a basis if it is independent and G=
1 s
(cid:10)e (cid:11)(cid:2)···(cid:2)(cid:10)e (cid:11).
1 s
Wewritesequencesmultiplicativelyandconsiderthemaselementsofthefreeabelian
monoidoverG,apointofviewwhichwasputforwardbytherequirementsofthetheory
of non-unique factorizations. Thus, we have at our disposal all notions from elementary
divisibility theory which provides a suitable framework when dealing with subsequences
ofgivensequences,andwemayapplyalgebraicconceptsinanaturalway.
LetF(G)bethefreeabelianmonoid,multiplicativelywritten,withbasisG.Theelements
ofF(G)arecalledsequencesoverG.WewritesequencesS ∈F(G)intheform
(cid:2)
S= gvg(S) with v (S)∈N for all g ∈G.
g 0
g∈G
We call v (S) the multiplicity of g in S, and we say that S contains g, if v (S)>0. S is
g g
called squarefree if v (S)(cid:2)1 for all g ∈ G. The unit element 1 ∈ F(G) is called the
g
emptysequence.AsequenceS iscalledasubsequenceofSifS |SinF(G)(equivalently,
1 1
v (S )(cid:2)v (S)forallg ∈G),anditiscalledapropersubsequenceofSifitisasubsequence
g 1 g
with1(cid:9)=S (cid:9)=S.IfasequenceS ∈F(G)iswrittenintheformS=g ·...·g,wetacitly
1 1 l
assumethatl ∈N andg ,...,g ∈G.
0 1 l
340 W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369
Forasequence
(cid:2)
S=g ·...·g = gvg(S) ∈F(G),
1 l
g∈G
wecall
(cid:1)
|S|=l= v (S)∈N the length of S,
g 0
g∈G
h(S)=max{v (S)|g ∈G}∈[0,|S|] the maximum of the multiplicities of S,
g
(cid:1)l
1
k(S)= ∈Q(cid:1)0 the cross-number of S,
ord(g )
i=1 i
supp(S)={g ∈G|v (S)>0}⊂G the support of S,
g
(cid:1)l (cid:1)
(cid:1)(S)= g = v (S)g ∈G the sum of S,
i g
i=1 g∈G
(cid:3) (cid:4) (cid:5)
(cid:1) (cid:4)
(cid:4)
(cid:2) (S)= g (cid:4)I ⊂[1,l] with |I|=k
k i(cid:4)
i∈I
the set of k-term subsums of S, for all k ∈N,
(cid:6) (cid:6)
(cid:2)(cid:2)k(S)= (cid:2)j(S), (cid:2)(cid:1)k(S)= (cid:2)j(S),
j∈[1,k] j(cid:1)k
and
(cid:2)(S)=(cid:2)(cid:1)1(S) the set of (all) subsums of S.
ThesequenceS iscalled
• zero-sumfreeif0∈/(cid:2)(S),
• azero-sumsequenceif(cid:1)(S)=0,
• aminimalzero-sumsequenceifitisanon-emptyzero-sumsequenceandeveryproper
subsequenceiszero-sumfree,
• ashortzero-sumsequenceifitisazero-sumsequenceoflength|S|∈[1,exp(G)].
WedenotebyB(G)thesetofallzero-sumsequencesandbyA(G)thesetofallminimal
zero-sumsequences.ThenB(G) ⊂ F(G)isasubmonoid(alsocalledtheblockmonoid
overG);itisaKrullmonoidandA(G)isthesetofatomsofB(G)(see[96,Proposition
2.5.6]).Foranymapofabeliangroups(cid:3):G → G(cid:13),thereexistsauniquehomomorphism
(cid:3):F(G) → F(G(cid:13))with(cid:3)|G=(cid:3).Usually,wesimplywrite(cid:3)insteadof(cid:3).Explicitly,
(cid:3):F(G) → F(G(cid:13)) is given by (cid:3)(g · ... · g) = (cid:3)(g ) · ... · (cid:3)(g) for all l ∈ N
1 l 1 l 0
and g ,...,g ∈ G. If S ∈ F(G), then |(cid:3)(S)|=|S| and supp((cid:3)(S))=(cid:3)(supp(S)). If
1 l
(cid:3):G → G(cid:13) isevenahomomorphism,then(cid:1)((cid:3)(S))=(cid:3)((cid:1)(S)),(cid:2)((cid:3)(S))=(cid:3)((cid:2)(S))and
W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369 341
(cid:3)(B(G)) ⊂ B(G(cid:13)). In particular, we use the inversion (g (cid:14)→ −g) and the translation
(g (cid:14)→g +g),andforS=g ·...·g ∈F(G)weset
0 1 l
−S=(−g )·...·(−g) and g +S=(g +g )·...·(g +g)∈F(G).
1 l 0 0 1 0 l
Ifg ∈Gisanon-zeroelementand
S=(n g)·...·(ng), where l ∈N and n ,...,n ∈[1,ord(g)],
1 l 0 1 l
then
n +···+n
(cid:15)S(cid:15) = 1 l
g
ord(g)
iscalledtheg-norm ofS.IfS isazero-sumsequenceforwhich{0}(cid:9)=(cid:10)supp(S)(cid:11)⊂Gis
afinitecyclicgroup,then
ind(S)=min{(cid:15)S(cid:15) |g ∈G with (cid:10)supp(S)(cid:11)=(cid:10)g(cid:11)}∈N
g 0
iscalledtheindexofS.Wesetind(1)=0,andifsupp(S)={0},thenwesetind(S)=1.
Nextwegivethedefinitionofthezero-suminvariantswhichwearegoingtodiscussin
thesubsequentsections.Weconcentrateoninvariantsdealingwithgeneralsequences,as
introduced in Definition 2.1. However, by an often used technique, problems on general
sequencesarereducedtoproblemsonsquarefreesequences,andthuswebrieflydealalso
with invariants on squarefree sequences (or in other words, with sets), as introduced in
Definition2.2.
Definition2.1. Letexp(G)=nandk,m∈Nwithk(cid:1)exp(G).Wedenoteby
• D(G)thesmallestintegerl ∈ NsuchthateverysequenceS ∈ F(G)oflength|S|(cid:1)l
hasanon-emptyzero-sumsubsequence.Theinvariant D(G) iscalledtheDavenport
constant ofG.
• d(G)themaximallengthofazero-sumfreesequenceoverG.
• (cid:4)(G) the smallest integer l ∈ N such that every sequence S ∈ F(G) of length |S|(cid:1)l
hasashortzero-sumsubsequence.
• s (G)thesmallestintegerl ∈NsuchthateverysequenceS ∈F(G)oflength|S|(cid:1)l
mn
hasazero-sumsubsequenceT oflength|T|=mn.Inparticular,wesets(G)=s (G).
n
• snN(G)thesmallestintegerl ∈NsuchthateverysequenceS ∈F(G)oflength|S|(cid:1)l
hasanon-emptyzero-sumsubsequenceT oflength|T|≡0modn.
• E (G)thesmallestintegerl ∈ NsuchthateverysequenceS ∈ F(G)oflength|S|(cid:1)l
k
hasazero-sumsubsequenceT withk(cid:1)|T|.
• (cid:5)(G)thesmallestintegerl ∈N withthefollowingproperty:
0
Foreveryzero-sumfreesequenceS ∈F(G)oflength|S|(cid:1)lthereexistasubgroup
H ⊂Gandanelementa ∈G\H suchthat G•\(cid:2)(S)⊂a+H.
Asimpleargument(see[96,Section5.1]fordetails)showsthat
d(G)=max{|S||S ∈F(G), (cid:2)(S)=G•} and
1+d(G)=D(G)=max{|S||S ∈A(G)}.
342 W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369
Definition2.2. Wedenoteby
• Ol(G) the smallest integer l ∈ N such that every squarefree sequence S ∈ F(G) of
length|S|(cid:1)lhasanon-emptyzero-sumsubsequence.TheinvariantOl(G)iscalledthe
OlsonconstantofG.
• ol(G)themaximallengthofasquarefreezero-sumfreesequenceS ∈F(G).
• cr(G) the smallest integer l ∈ N such that every squarefree sequence S ∈ F(G•) of
length|S|(cid:1)lsatisfies(cid:2)(S)=G.Theinvariantcr(G) iscalledthecriticalnumberofG.
• g(G)thesmallestintegerl ∈NsuchthateverysquarefreesequenceS ∈F(G)oflength
|S|(cid:1)lhasazero-sumsubsequenceT oflength|T|=exp(G).
Weusetheconventionthatmin(∅)=sup(∅)=0.ForasubsetG ⊂Gandsomeinteger
0
l ∈N,R.B.EggletonandP.Erdo˝s(see[41])introducedthef-invariant
f(G ,l)=min{|(cid:2)(S)||S ∈F(G ), S squarefree and zero-sumfree, |S|=l}.
0 0
ThebasicrelationshipsbetweentheseinvariantsaresummarizedinLemma10.1.
3. On the Davenport constant
LetG=C (cid:2)···(cid:2)C with1<n |...|n ,r =r(G)andlet(e ,...,e )beabasis
n1 nr 1 r 1 r
ofGwithord(e )=n foralli ∈[1,r].Thenthesequence
i i
(cid:2)r
S= eni−1 ∈F(G)
i
i=1
iszero-sumfreewhencewehavethecrucialinequality
d(G)(cid:1)d∗(G).
Inthe1960s,D.KruyswijkandJ.E.Olsonprovedindependentlythefollowingresult(see
[143,5,44,144]and[96,Theorems5.5.9and5.8.3]).
Theorem3.1. IfGisap-grouporr(G)(cid:2)2,thend(G)=d∗(G).
We present two types of results implying that d(G)=d∗(G). The first one is due to
P. van Emde Boas et al. (see [44, Theorems 3.9, 4.2], where more results of this flavor
maybefound)andthesecondisduetoS.T.Chapmanetal.(see[25],andalsothevarious
conjecturesinthatpaper).
Theorem3.2. LetG=C (cid:2)C (cid:2)C andH=C (cid:2)C (cid:2)C with1(cid:2)n |n |n .
If(cid:5)(H)=d∗(H)−1,the2nn1d(G)2=n2d∗(G2n)3. n1 n2 n3 1 2 3
Theorem3.3. LetG=H(cid:2)C wherek,m∈NandH ⊂Gisasubgroupwithexp(H)|m.
km
Ifd(H(cid:2)C )=d(H)+m−1and(cid:4)(H(cid:2)C )(cid:2)d(H)+2m,thend(G)=d(H)+km−1.
m m
W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369 343
In particular (use Theorem 3.1 and [96, Proposition 5.7.7]), if m is a prime power and
d(H)<m,thend(G)=d∗(G).
These and similar results give rise to long lists of explicit groups satisfying d(G) =
∗ ∗
d (G)(see[44,25,6,46,35]).ThefirstexampleofagroupGwithd(G)>d (G)isdueto
P.C.Baayen.In[44,Theorem8.1]itisshownthat
d(G)>d∗(G) for G=C24k(cid:2)C4k+2 with k ∈N,
andmoreexamplesaregivenin[46].LetH ⊂Gbeasubgroup.Thend(H) + d(G/H)(cid:2)
d(G),andifGisasabove,I ⊂[1,r]and
(cid:7)
H = C then d(H)>d∗(H) implies d(G)>d∗(G)
ni
i∈I
∗
(see [96, Proposition 5.1.11]).This shows that the interesting groups with d(G)>d (G)
arethosewithsmallrank.A.GeroldingerandR.Schneidershowedthatthereareinfinitely
manyGwithr(G)=4suchthatd(G)>d∗(G).Thefollowingresultmaybefoundin[98]
and[77,Theorem3.3].
∗
Theorem3.4. Wehaved(G)>d (G)ineachofthefollowingcases:
1. G=Cm(cid:2)Cn2(cid:2)C2n,wherem,n∈N(cid:1)3areoddandm|n.
2. G=C2i(cid:2)C25n−i,wheren∈N(cid:1)3isoddandi ∈[2,4].
LetG=C2r(cid:2)Cn wherer ∈Nandn∈N(cid:1)3 isodd.Thend(G)=d∗(G)ifandonlyif
r(cid:2)4(see[98,Corollary2]).Forsomesmallr(cid:1)5andn(cid:1)3theprecisevalueofd(G)was
recentlydeterminedin[49].Thegrowthofd(G)−d∗(G)isstudiedin[140].
Wemakethefollowingconjecture.
Conjecture3.5. Ifr(G)=3orG=Cnr withn,r ∈N(cid:1)3,thend(G)=d∗(G).
ForgroupsofrankthreeConjecture3.5goesbacktoP.vanEmdeBoas(see[46])andis
supportedby[69].ForgroupsoftheformG=Cr itissupportedby[80,Theorem6.6].
n
ThenextresultprovidesupperboundsonD(G).ThefirstoneisduetoP.vanEmdeBoas
andD.Kruyswijk[46,Theorem7.1]andissharpforcyclicgroups(forotherapproaches
andrelatedboundssee[8,141]).Thesecondboundissharpforgroupsofrank2andwith
H =pG for some prime divisor p of exp(G) (see [96, Theorem 5.5.5 and Proposition
5.7.11]).
Theorem3.6. Letexp(G)=n(cid:1)2andH ⊂Gbeasubgroup.
1. d(G)(cid:2)(n−1)+nlog|G|.
n
2. d(G)(cid:2)d(H)exp(G/H)+max{d(G/H),(cid:4)(G/H)−exp(G/H)−1}.
Weendthissectionwithaconjecturesupportedby[96,Theorem6.2.8].
344 W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369
Conjecture3.7. If|G|>1,thenD(G)(cid:2)d∗(G)+r(G).
4. On the structure of long zero-sumfree sequences
LetS ∈ F(G)beazero-sumfreesequenceoflength|S|=d(G).Accordingtogeneral
philosophyininverseadditivenumbertheory(see[142,53,54]),Sshouldhavesomestruc-
ture. Obviously, if G is cyclic of order n(cid:1)2, then S =gn−1 for some g ∈ supp(S) with
ord(g)=n,andifSisanelementary2-groupofrankr,thenS=e ·...·e forsomebasis
1 r
(e ,...,e )ofG.Apartfromthesetrivialcasesverylittleisknownuptonow.Themost
1 r
modestquestionsonecouldaskarethefollowing:
1. Whatistheorderofelementsinsupp(S)?
2. Whatisthemultiplicityofelementsinsupp(S)?Whatisareasonablelowerboundfor
h(S)?
3. Howlargeis supp(S)?
Crucial in all investigations of zero-sumfree sequences is the following inequality of
Moser-Scherk(see[96,Theorem5.3.1]):LetS ∈F(G)beazero-sumfreesequence.
If S=S S then |(cid:2)(S)|(cid:1)|(cid:2)(S )|+|(cid:2)(S )|.
1 2 1 2
ByM.FreezeandW.W.Smith([52,Theorem2.5,96,Proposition5.3.5])thisimpliesthat
|(cid:2)(S)|(cid:1)2|S|−h(S)(cid:1)|S|+|supp(S)|−1.
Westartwiththefollowingconjecture.
Conjecture4.1. Everyzero-sumfreesequenceS ∈ F(G)oflength|S|=d(G)hassome
elementg ∈supp(S)withord(g)=exp(G).
Theconjectureistrueforcyclicgroups,p-groups(see[96,Corollary5.1.13]),groupsof
theformG=C (cid:2)C (seebelow)andforG=C (cid:2)C (see[78]).Asconcernsthesecond
n n 2 2n
question,thephilosophyisthatingroupswheretheexponentislargeincomparisonwith
therank,h(S)shouldbelarge.
For cyclic groups, there are the following results going back to J.D. Bovey, P. Erdo˝s,
I.Niven,W.Gao,A.GeroldingerandY.ouldHamidoune(see[17,76,97]and[96,Theorem
5.4.5]).
Theorem4.2. LetGbecyclicofordern(cid:1)3,andletS ∈F(G)beazero-sumfreesequence
oflength
n+1
|S|(cid:1) .
2
1. Forallg ∈supp(S)wehaveord(g)(cid:1)3.
2. Thereexistssomeg ∈supp(S)withv (S)(cid:1)2|S|−n+1.
g
W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369 345
3. Thereexistssomeg ∈supp(S)withord(g)=nsuchthat
n+5
v (S)(cid:1) if n is odd and v (S)(cid:1)3 if n iseven.
g g
6
Incyclicgroupslongzero-sumfreesequencesandlongminimalzero-sumsequencescan
becompletelycharacterized(see[71]).
Theorem4.3. LetGbycyclicofordern(cid:1)2andletS ∈F(G)beazero-sumfreesequence
oflength|S|=n−kwithk ∈[1,(cid:5)n/3(cid:6)]+1].Thenthereexistssomeg ∈Gwithord(g)=n
andx1,...,xk−1 ∈[1,n−1]suchthat
k(cid:2)−1 (cid:1)k−1
S=gn−2k+1 (x g) and x (cid:2)2k−2.
i i
i=1 i=1
Inparticular,everyminimalzero-sumsequenceS ∈ A(G)oflength|S|(cid:1)n−(cid:5)n/3(cid:6)has
ind(S)=1.
The index of zero-sum sequences over cyclic groups is investigated in [71,26,29].
In [127] (p. 344 with d =n) it is conjectured that every sequence S ∈ F(C ) of length
n
|S|=n has a non-empty zero-sum subsequence T with ind(T)=1.Among others, the
g-normandtheindexofzero-sumsequencesplayaroleinarithmeticalinvestigations(see
[96,Section6.8]).
NextwediscussgroupsoftheformG=C (cid:2)C withn(cid:1)2(see[77,167,79],[96,Section
n n
5.8]and[131]).
Theorem4.4. LetG=C (cid:2)C withn(cid:1)2.Thenthefollowingstatementsareequivalent:
n n
(a) IfS ∈ F(G),|S|=3n−3andS hasnozero-sumsubsequenceT oflength|T|(cid:1)n,
thenthereexistssomea ∈Gsuchthat0n−1an−2|S.
(b) IfS ∈F(G)iszero-sumfreeand|S|=d(G),thenan−2|S forsomea ∈G.
(c) IfS ∈A(G)and|S|=D(G),thenan−1|S forsomea ∈G.
(d) If S ∈ A(G) and |S|=D(G), then there exists a basis (e ,e ) of G and integers
1 2
x ,...,x ∈[0,n−1]withx +···+x ≡1modnsuchthat
1 n 1 n
(cid:2)n
S=e1n−1 (x(cid:5)e1+e2).
(cid:5)=1
Moreover,ifS ∈A(G)and|S|=D(G),thenord(g)=nforeveryg ∈supp(S),andif
nisprime,then|supp(S)|∈[3,n].
Conjecture4.5. For every n(cid:1)2 the four equivalent statements of Theorem 4.4 are
satisfied.
346 W.Gao,A.Geroldinger/Expo.Math.24(2006)337–369
Conjecture4.5hasbeenverifiedforn∈[2,7],andifitholdsforsomen(cid:1)6,thenitholds
for2n(see[79,Theorem8.1]).Wecontinuewitharesultfornon-cyclicgroupshavinglarge
exponent(see[79]).
Theorem4.6.
1. LetG=C (cid:2)C with1<n |n andn >n (n +1).Let(cid:3):G→G=C (cid:2)C be
n1 n2 1 2 2 1 1 n1 n1
thecanonicalepimorphismandS ∈ A(G)oflength|S|=D(G).Ifgk|(cid:3)(S)forsome
k>n andsomeg ∈G,thengk|S forsomeg ∈(cid:3)−1(g).
1
2. LetG=H(cid:2)C whereexp(G)=n=lm,H ⊂ Gasubgroupwithexp(H)|m,m(cid:1)2
n
andl(cid:1)4|H|>4(m−2).Let(cid:3):G→G=H(cid:2)C denotethecanonicalepimorphism
m
andS ∈ F(G)azero-sumfreesequenceoflength|S|=n.ThenS hasasubsequence
T oflength|T|(cid:1)(l−2|H|+1)msuchthatthefollowingholds:Ifgk|(cid:3)(T)forsome
k>mandsomeg ∈G,thengk|T forsomeg ∈(cid:3)−1(g).
For general finite abelian groups there is the following result (see [76], [96, Theorem
5.3.6])whichplaysakeyroleintheproofofTheorem10.4.2).
Theorem4.7. LetG ⊂Gbeasubset,k ∈Nandk(cid:1)2besuchthatf(G ,k)>0,andlet
0 0
S ∈F(G )beazero-sumfreesequenceoflength
0
(cid:8) (cid:9)
|G|−k
|S|(cid:1) +1 k.
f(G ,k)
0
Thenthereexistssomeg ∈G suchthat
0
|S| |G|−k−1
v (S)(cid:1) − .
g k−1 (k−1)f(G ,k)
0
If the rank of the group is large in comparison with the exponent, there is in gen-
eral no element with high multiplicity (see Theorem 10.4.1), but in case of elementary
p-groupsthereisthefollowingstructuralresult(see[77,Theorem10.3],[80,Corollary6.3],
[96,Corollary5.6.9]).
Theorem4.8. Let G be a finite elementary p-group and S ∈ F(G) be a zero-sumfree
sequence of length |S|=d(G). Then (g,h) is independent for any two distinct elements
g,h∈supp(S).
Wecontinuewiththefollowing
Conjecture4.9. LetG = C (cid:2)···(cid:2)C with1<n |···|n ,k ∈ [1,n −1]andS ∈
F(G) be a sequence of lengnth1 |S|=k +nrd(G). If S h1as no zrero-sum sub1sequence S(cid:13) of
length|S(cid:13)|>k,thenS=0kT whereT ∈F(G)iszero-sumfree.