Inverse zero-sum problems and algebraic invariants

Transcription

1 Iverse zero-sum problems a algebraic ivariats Bejami Girar To cite this versio: Bejami Girar. Iverse zero-sum problems a algebraic ivariats. Acta Arithmetica, Istytut Matematyczy PAN, 2008, 135 (3, pp < /aa >. <hal v2> HAL I: hal Submitte o 18 Oct 2010 HAL is a multi-iscipliary ope access archive for the eposit a issemiatio of scietific research ocumets, whether they are publishe or ot. The ocumets may come from teachig a research istitutios i Frace or abroa, or from public or private research ceters. L archive ouverte pluriiscipliaire HAL, est estiée au épôt et à la iffusio e ocumets scietifiques e iveau recherche, publiés ou o, émaat es établissemets eseigemet et e recherche fraçais ou étragers, es laboratoires publics ou privés.

2 INVERSE ZERO-SUM PROBLEMS AND ALGEBRAIC INVARIANTS by Bejami Girar Abstract. I this article, we stuy the maximal cross umber of log zero-sumfree sequeces i a fiite Abelia group. Regarig this iverse-type problem, we formulate a geeral cojecture a prove, amog other results, that this cojecture hols true for fiite cyclic groups, fiite Abelia p-groups a for fiite Abelia groups of rak two. Also, the results obtaie here eable us to improve, via the resolutio of a liear iteger program, a result of W. Gao a A. Geroliger cocerig the miimal umber of elemets with maximal orer i a log zero-sumfree sequece of a fiite Abelia group of rak two. 1. Itrouctio Let G be a fiite Abelia group, writte aitively. By exp(g we eote the expoet of G. If G is cyclic of orer, it will be eote by C. I the geeral case, we ca ecompose G (see for istace [23] as a irect prouct of cyclic groups C 1 C r where 1 < 1... r N. I this paper, ay fiite sequece S = (g 1,...,g l of l elemets from G will be calle a sequece i G with legth S = l. Give a sequece S = (g 1,...,g l i G, we say that s G is a subsum of S whe it lies i the followig set, { } Σ(S = g i I {1,...,l}. i I If 0 is ot a subsum of S, we say that S is a zero-sumfree sequece. If l g i = 0, the S is sai to be a zero-sum sequece. If moreover oe has i I g i 0 for all proper subsets I {1,...,l}, S is calle a miimal zero-sum sequece. I a fiite Abelia group G, the orer of a elemet g will be writte or(g a for every ivisor of the expoet of G, we eote by G the subgroup of G cosistig of all elemets of orer iviig : G = {x G x = 0}. For every ivisor of exp(g, a every sequece S i G, we eote by α the umber of elemets, coute with multiplicity, cotaie i S a of orer. Although the quatity Mathematics Subject Classificatio (2000: 11R27, 11B75, 11P99, 20D60, 20K01, 05E99, 13F05.

3 α clearly epes o S, we will ot emphasize this epeece i the preset paper, sice there will be o risk of cofusio. Let P be the set of prime umbers. Give a positive iteger N = N\{0}, we eote by D the set of its positive ivisors a we set τ( = D. If > 1, we eote by P ( the smallest prime elemet of D, a we put by covetio P (1 = 1. For every prime p P, ν p ( will eote the p-aic valuatio of. Let G C 1 C r, with 1 < 1... r N, be a fiite Abelia group. We set: r D (G = ( i 1+1 as well as (G = D (G 1. By D(G we eote the smallest iteger t N such that every sequece S i G with S t cotais a o-empty zero-sum subsequece. The umber D(G is calle the Daveport costat of the group G. By(Gweeotethelargestitegert N suchthatthereexistsazero-sumfreesequece S i G with S = t. It ca be reaily see that, for every fiite Abelia group G, oe has (G = D(G 1. If G C ν1 C νs, with ν i > 1 for all i 1,s, is the logest possible ecompositio of G ito a irect prouct of cyclic groups, the we set: s k ν i 1 (G =. ν i The cross umber of a sequece S = (g 1,...,g l, eote by k(s, is the efie by: l 1 k(s = or(g i. The otio of cross umber was itrouce by U. Krause i [17] (see also [18]. Fially, we efie the so-calle little cross umber k(g of G: k(g = max{k(s S zero-sumfree sequece of G}. Give a fiite Abelia group G, two elemetary costructios (see [11], Propositio give the followig lower bous: D (G D(G a k (G k(g. The ivariats D(G a k(g play a key rôle i the theory of o-uique factorizatio (see for istace Chapter 9 i [20], the book [11] which presets the ifferet aspects of the theory, a the survey [12] also. They have bee extesively stuie urig last ecaes a eve if umerous results were prove (see Chapter 5 of the book [11], [7] for a survey with may refereces o the subject, a [14] for recet results o the cross umber of fiite Abelia groups, their exact values are kow for very special types of groupsoly. Ithesequel, we will ee some ofthese values ithe case offiite Abeliapgroups a fiite Abelia groups of rak two, so we gather them i the followig theorem (see [10], [21] a [22]. 2

4 Theorem 1.1. The followig two statemets hol. (i Let p P, r N a a 1 a r, where a i N for all i 1,r. The, for the p-group G C p a 1 C p ar, we have: r r ( D(G = (p a i p 1+1 = D a i 1 (G a k(g = = k (G. (ii For every m, N, we have: p a i D(C m C = m+ 1 = D (C m C. I particular, we have D(C =. The aim of this paper is to stuy some iverse zero-sum problems of a special type. Istea of tryig to characterize explicitly, give a fiite Abelia group, the structure of log zero-sumfree sequeces (see [5], [3], [9], [25] a [8], or the structure of zero-sumfree sequeces with large cross umber (see [13], we stuy to what extet a zero-sumfree sequece ca be extremal i both irectios simultaeously. For istace, what is the maximal cross umber of a log zero-sumfree sequece? Regarig this problem, we propose the followig geeral cojecture. Cojecture 1.2. Let G C 1 C r, with 1 < 1... r N, be a fiite Abelia group. Give a zero-sumfree sequece S i G verifyig S (G, oe always has the followig iequality: r ( i 1 k(s. I particular, oe has k(s < r. Oe ca otice that Cojecture 1.2 is closely relate to the istributio of the orers of elemets i a log zero-sumfree sequece. As we will see i this paper, it provies, whe it hols, useful iformatios o this questio. I the followig propositio, we gather what is curretly kow, to the best of our kowlege, o the structure of log zerosumfree sequeces i fiite Abelia groups of rak two. This result, ue to W. Gao a A. Geroliger, ca be fou uer a slightly ifferet form i [11], Propositio Propositio 1.3. Let G C m C, where m, N, be a fiite Abelia group of rak two. For every zero-sumfree sequece S i G with S = (G = m+ 2, the followig two statemets hol. (i For every elemet g S, oe has m or(g. (ii The sequece S cotais at least ( 2m 2 m+ P ( +1 1 m 1 elemets with orer. The problem of the exact structure of a log zero-sumfree sequece i groups of the form G C m C is also closely relate to a importat cojecture i aitive group theory, which bears upo the so-calle Property B. Let 2 be a iteger. We say that i 3

5 has Property B if every zero-sumfree sequece i G C C with S = (G = 2 2 cotais some elemet repeate at least 2 times. Property B was itrouce a first stuie i [4] (see also [11], Sectio 5.8, [19] a [9]. It is cojecture that every iteger 2 has Property B, a recetly, it was prove that a iteger 2 has Property B if each prime ivisor of has this property (see [6], Sectio 8 a [8]. Therefore, it remais to solve this problem for prime umbers. Regarig this, it ca be show that Property B hols for = 2,3,5,7 (see [6], Propositio 4.2, for = 11,13,17,19(see [1], a cosequetly for every iteger beig represetable as a prouct of these umbers. Moreover, W. Schmi prove i [25] that if some iteger m 2 has Property B, the the zero-sumfree sequeces i G C m C with legth (G = m + 2 ca be characterize explicitly for all N. This result provies a uifie way to prove Theorem 3.3 i [5] a Theorem i [3]. It also implies, assumig that Property B hols for every iteger 2, that Cojecture 1.2 hols true for every fiite Abelia group of rak two. 2. New results a pla of the paper I this article, we prove that Cojecture 1.2 hols for several types of fiite Abelia groups. To begi with, i Sectio 3, we prove some cosequeces of this cojecture i the cases where it hols. For istace, Cojecture 1.2, if true, woul imply simultaeously two classical a log-staig cojectures relate to the Daveport costat of fiite Abelia groups of the form C r. Propositio 2.1. Let,r N be such that Cojecture 1.2 hols for the group C r. The, oe has the followig equality: D(C r = r( 1+1. Moreover, every zero-sumfree sequece S i C r with S = (Cr = r( 1 cosists oly of elemets with orer. More geerally, Cojecture 1.2, if true, woul provie the followig geeral upper bou for the Daveport costat of a fiite Abelia group. Propositio 2.2. Suppose that Cojecture 1.2 hols for G C 1 C r, with 1 < 1... r N. The, oe has the followig iequality: r r r ( D(G ( i 1+1 = D r (G+ 1 ( i 1. i i The, i Sectio 3 also, we prove that Cojecture 1.2 hols true for fiite cyclic groups a fiite Abelia p-groups. Propositio 2.3. Cojecture 1.2 hols for the followig groups G. (i G is a fiite cyclic group. (ii G is a fiite Abelia p-group. 4

6 I Sectio 4, we preset a geeral metho which was itrouce i [14] so as to stuy the cross umber of fiite Abelia groups. The, usig this metho, we prove i Sectio 5 two importat lemmas, which will be useful i the stuy of the special case of fiite Abelia groups of rak two. I Sectio 6, we prove the two mai theorems of this paper. The first oe states that Cojecture 1.2 hols for every fiite Abelia group of rak two. As alreay metioe i Sectio 1, this result supports Property B (see [25]. Theorem 2.4. Let G C m C, where m, N, be a fiite Abelia group of rak two. For every zero-sumfree sequece S i G with S (G = m+ 2, the followig iequality hols: ( ( m 1 1 k(s +. m I particular, oe always has k(s < 2. The seco theorem, which is prove i Sectio 6 as well, is a effective result which states that, i a fiite Abelia group of rak two, most of the elemets of a log zerosumfree sequece must have maximal orer. This result improves sigificatly the statemet of Propositio 1.3 (ii. Theorem 2.5. Let G C m C, where m, N, be a fiite Abelia group of rak two. For every zero-sumfree sequece S i G with S = (G = m+ 2, the followig two statemets hol. (i If is a prime power, the S cotais at least 1 elemets with orer. (ii If is ot a prime power, the S cotais at least 4 ( elemets with orer. It may be observe that for every group G C m C, where m, N a 2, there exists a zero-sumfree sequece S i G with S = (G = m+ 2, a which oes ot cotai strictly more tha 1 elemets with orer. Iee, let (e 1,e 2 be a basis of G, with or(e 1 = m a or(e 2 =. The, it suffices to cosier the zero-sumfree sequece S cosistig ofthe elemet e 1 repeate m 1times athe elemet e 2 repeate 1 times. From this poit of view, Theorem 2.5 proves to be early optimal. I aitio, the geeral metho presete i Sectio 4 ca be successfully use to prove a aalogue of Theorem 2.5 i the case of fiite Abelia p-groups (see [15]. Fially, i Sectio 7, we will preset a iscuss a geeral cojecture cocerig the maximal possible legth of a zero-sumfree sequece with large cross umber, which ca be see as a ual versio of Cojecture

7 3. Proofs of Propositios 2.1, 2.2 a 2.3 To start with, we prove the two corollaries of Cojecture 1.2 aouce i Sectio 2. Proof of Propositio 2.1. Let S be a zero-sumfree sequece i G C r with maximal legth S = (G = D(G 1. The, oe has the followig iequality: D(G 1 = S ( 1 k(s r, which implies that D(G r( 1+1 = D (G, a sice D (G D(G always hols, the equality follows. Cosequetly, oe has: ( 1 k(s = r = D(G 1 = S, a so, every elemet g of S verifies or(g = exp(g =. Proof of Propositio 2.2. Let S be a zero-sumfree sequece i G C 1 C r, with 1 < 1... r N, such that S = (G = D(G 1. The, oe has the followig iequality: D(G 1 r which implies the esire result. = S r k(s r ( i 1 We prove ow that Cojecture 1.2 hols true for fiite cyclic groups a fiite Abelia p-groups. Proof of Propositio 2.3. (i Let 2 be a iteger a let S be a zero-sumfree sequece i C with S (C = 1. The, it is well-kow (see for istace [11], Theorem (i that there exists g C with or(g = such that S is of the followig form: S = (g,...,g. }{{} 1 times Cosequetly, we obtai: which gives the esire result. k(s = 1, (ii Let p P, r N, a G C p a 1 C p ar, with a 1 a r a a i N for all i 1,r, be a p-group. By Theorem 1.1 (i, oe has: r ( p a i 1 k(g = = k (G. p a i The, for every zero-sumfree sequece S i G, i particular for those verifyig S (G, oe iee has, by the very efiitio of the little cross umber: r ( p a i 1 k(s k(g =, a the proof is complete. p a i i, 6

8 4. Outlie of a ew metho Let G be a fiite Abelia group, a let S be a sequece of elemets i G. The geeral metho that we will use i this paper (see also [14] a [15] for applicatios of this metho i two other cotexts, cosists i cosierig, for every, N such that 1 exp(g, the followig exact sequece: π (, 0 G / G G G / 0. Now, let U be the subsequece of S cosistig of all the elemets whose orer ivies. If, for some 1 exp(g, it is possible to fi sufficietly may isjoit o-empty zero-sum subsequeces i π (,(U, that is to say sufficietly may isjoit subsequeces iu thesum ofwhich areelemets of oreriviig /, thes caot beazero-sumfree sequece i G. So as to make this iea more precise, we propose i [14] to itrouce the followig umber, which ca be see as a extesio of the classical Daveport costat. Let G C 1 C r, with 1 < 1... r N, be a fiite Abelia group a, N be two itegers such that 1 exp(g. By D (,(G we eote the smallest iteger t N such that every sequece S i G with S t cotais a subsequece with sum i G /. Usig this efiitio, we ca prove the followig simple lemma, which is oe possible illustratio of the iea we presete. This result will be useful i Sectio 5 a states that give a fiite Abelia group G, there exist strog costraits o the way the orers of elemets have to be istribute withi a zero-sumfree sequece. Lemma 4.1. Let G be a fiite Abelia group a, N be two itegers such that 1 exp(g. Give a sequece S of elemets i G, we will write T for the subsequece of S cosistig of all the elemets whose orer ivies /, a we will write U for the subsequece of S cosistig of all the elemets whose orer ivies (I particular, oe has T U. The, the followig coitio implies that S caot be a zero-sumfree sequece: U T T + D D (,(G (, (G. Proof. Letusset = D (, (G. Wheithols, thisiequality implies thatthereare isjoit subsequeces S 1,...,S of S, the sum of which are elemets of orer iviig /. Now, by the very efiitio of D (, (G, S has to cotai a o-empty zero-sum subsequece. Now, i orer to obtai effective iequalities from the symbolic costraits of Lemma 4.1,oecausearesultprovei[14],whichstatesthatforayfiiteAbeliagroupGa every 1 exp(g, the ivariat D (,(G is like with the classical Daveport costat of a particular subgroup of G, which ca be characterize explicitly. I orer to efie properly this particular subgroup, we have to itrouce the followig otatio. 7

9 For all i 1,r, we set: A i = gc(, i, B i = lcm(, i lcm(, i a υ i (, = A i gc(a i,b i. For istace, wheever ivies i, we have υ i (, = gc(, i =, a i particular υ r (, =. We ca ow state our result o D (,(G (see [14], Propositio 3.1. Propositio 4.2. Let G C 1 C r, with 1 < 1... r N, be a fiite Abelia group a, N be such that 1 exp(g. The, we have the followig equality: D (,(G = D ( C υ1 (, C υr(,. 5. Two lemmas relate to zero-freeess i G C m C I this sectio, we show how the metho presete i Sectio 4 ca be use i orer to obtai two key lemmas for the proofs of Theorems 2.4 a 2.5. To start with, we prove the followig result. Lemma 5.1. Let G C m C, where m, N, 2, be a fiite Abelia group of rak two, a let S be a zero-sumfree sequece i G with S (G = m+ 2. The, for every l D \{}, oe has the followig iequality: D l α m m 1. Proof. Let S be a zero-sumfree sequece i G C m C with S (G = m+ 2. Let l D \{}, = /l a =, which leas to / = ml. We also set m = gc(,m. Now, let T a U be the two subsequeces of S which are efie i Lemma 4.1. I particular, oe has T U = S, a by Propositio 1.3 (i, we obtai: T = Dlα m. To start with, we etermie the exact value of D (,(G. Oe has: υ 1 (, = m ( gc m, lcm(,m lcm(,m m = gc ( m, m m m = gc(m,m l = 1, a, sice υ 2 (, =, oe obtais, usig Propositio 4.2 a Theorem 1.1 (ii, the followig equalities: 8

10 D (,(G = D ( C υ1 (, C υ2 (, = D ( C l = l. Now, let us suppose that oe has T m. Sice l D \{}, we obtai the followig iequalities: T + U T D (,(G T + l(m+ 2 T m+ l( 2 = (m+ml 1 l ( l + > (m+ml 1 l 1 = D (, (G D (,(G, a, accorig to Lemma 4.1, S must cotai a o-empty zero-sum subsequece, which is a cotraictio. Thus, oe has T m 1, which is the esire result. Now, let 2beaiteger, ap 1,...,p r beitsistict primeivisors. Give m N a a zero-sumfree sequece S i G C m C with S (G = m+ 2, Lemma 5.1 implies that the itegers α m N, where D \{}, have to satisfy the followig r liear costraits: D /pi α m m 1, for all i 1,r. I the ext lemma, we solve a liear iteger program o the ivisor lattice of, i orer to obtai the maximum value of the fuctio (α m D\{} α m D \{} uer the r above costraits (the reaer itereste by liear programmig methos is referre to the book [26], for a exhaustive presetatio of the subject. Lemma 5.2. Let m, N, with 2, a let (x D\{} be a sequece of positive itegers, such that for every prime ivisor p of, oe has the followig liear costrait: D /p x m 1. The, oe has the followig iequality, which is best possible: x m 1. D \{} 9

11 Proof. Let 2 be a iteger, a p 1,...,p r be its istict prime ivisors. For every k 0,m 1, letalsos k bethesetofallthesequecesofpositiveitegersx = (x D\{} which verify the above liear costraits, a beig such that x 1 = m k 1. Now, we ca prove, by iuctio o k 0,m 1, that the followig statemet hols. x For every sequece x S k, oe has m 1. D \{} If k = 0, the for every x S 0, the liear costraits imply that x = 0 for all D \{1,}, which gives the followig equality: x = m 1. D \{} Assume ow that the statemet is vali for k 1 0. Let us efie the followig map: f : D \{} {A A 1,r } {i 1,r D /pi }. Let x S k a let L be the set of the elemets D \{1,} such that oe has x 1. By efiitio, a for every D \{}, f( is the umber of liear costraits i which the variable x appears. Thus, for every prime ivisor p of, x /p appears i oly oe liear costrait, a we may assume, without loss of geerality, that we have: D /p x = m 1. Hece, for every i 1,r, the set L D /pi is o-empty, a oe obtais: f( = 1,r. L Let us cosier a o-empty subset L of L verifyig the followig equality: L f( = 1,r, a beig of miimal cariality regarig this property. Sice f( is a o-empty set for every D \{}, the followig property has to hol: L f( 1,r for all L L. Now, oe ca otice the followig two facts. Fact 1. For every L, oe has f( r L +1, a i particular, L r. This fact is a straightforwar cosequece of the followig combiatorial lemma. Lemma 5.3. Let r N a A 1,,A s be s o-empty subsets of 1,r verifyig i = 1,r, a i 1,s A A i 1,r for every subset I 1,s. i I The, for all i 1,s, oe has the followig iequality: A i r s+1. 10

12 Proof. By symmetry, it suffices to prove that oe has A 1 r s+1. Assume to the cotrary that A 1 r s+2. Sice, for all i 1,s 1, the set A i+1 must cotai at least oe elemet from 1,r \(A 1 A i, oe obtais the followig iequality: A 1 A i+1 A 1 A i +1. Therefore, we euce by a easy iuctio argumet that oe has A 1 A s 1 (r s+2+(s 2 = r, a so A 1 A s 1 = 1,r, which is a cotraictio. Fact 2. For every D \{}, oe has the followig iequalities: mif 1 (f( i 1,r \f( 2 r f(. p νp i ( i Now, usig Facts 1 a 2, we ca prove the esire result, by cosierig the sequece y = (y D\{} obtaie from x i the followig way: x 1 +1 if = 1, y = x 1 if L, otherwise. x It is reaily see that y S k 1. Therefore, Facts 1 a 2 give the followig iequalities: m 1 which completes the proof. = D \{} y D \{} D \{} D \{} D \{} D \{} x x x x x, ( ( ( 1 1 L r f( L L 1 L ( 1 L 2 L 1 11

13 6. Proofs of the two mai theorems To start with, we show that every fiite Abelia group of rak two satisfies Cojecture 1.2. The followig proof of Theorem 2.4 cosists i a irect applicatio of Lemmas 5.1 a 5.2. Proof of Theorem 2.4. Let G C m C, where m, N, be a fiite Abelia group of rak two, a let S be a zero-sumfree sequece i G with S (G = m+ 2. Sice, by Theorem 1.1 (ii, oe has (G = (G, we obtai that S = (G = m + 2. If = 1, the the esire result follows irectly from Propositio 1.3 (i, sice every elemet of S has orer m. Now, let us suppose that 2. Usig Propositio 1.3 (i, we obtai: k(s = a we ca istiguish two cases. D α = D α m m, Case 1. α 1. I this case, applyig Propositio 1.3 (i, oe obtais: α m = S α, D \{} which implies the followig iequalities: k(s = = D \{} α m m + α + α + m + ( S α m ( S ( 1 ( m 1 m ( 1 ( 1 Case 2. α 1. The, by Lemmas 5.1 a 5.2, we obtai: α m k(s = m + α D \{} ( m 1 + α m ( ( m 1 1 +, m which completes the proof. Now, we prove Theorem 2.5, which gives a lower bou for the umber of elemets with maximal orer i a log zero-sumfree sequece of a fiite Abelia group of rak two.. 12

14 Proof of Theorem 2.5. Let G C m C, where m, N, be a fiite Abelia group of rak two, a let S be a zero-sumfree sequece i G with S = (G = m+ 2. (i Let p P a a N be such that = p a. If a = 0, the G C m C m a, by Propositio 1.3 (i, every elemet of S has orer m. Now, let us suppose that a 1, The, by Lemma 5.1, oe has: S α = which iee implies that D p a 1 m 1, α S (m 1 α m = m+ 2 (m 1 = 1. (ii If τ( 3, the has to be a prime power, a the esire result follows by (i. Now, let us suppose that D = { 0 = 1 < 1 < 2 < 3...} cotais at least four elemets. I particular, oe has 6. (1 (2 By Lemmas 5.1 a 5.2, oe has: that is 1 ( αm ( αm 2 D \{} Now, we ca istiguish two cases. α m m m 1 m, ( αm D > 3 Case 1. 3 = 4. The 1 = 2, 2 = 3 a (1 implies: ( αm ( αm ( αm α m D > 3 ( αm m 1 m. m 1 m. But sice α m = S α α m α m α m, D > 3 relatio (2 implies: ( m+ 2 α ( αm ( αm ( αm m 1 m, that is ( m+ 2 α 5 Now usig the fact that, by Lemma 5.1, oe has: ( αm +α ( m αm α m 2 3 m 1 m, α m 2 +α m 4 m 1 as well as α m 3 m 1, 13

15 we obtai ( ( ( m+ 2 α m 1 2(m m 1 m, which is equivalet to ( 1 α 5 m 1 m, that is a thus which is the esire result. 5( 1 (m 1 5α, 4 ( 5 + α, 5 5 (3 Case The (1 implies: ( αm ( αm D 3 α m m 1 m. But sice relatio (3 implies: ( m+ 2 α 5 α m = S α α m α m 1 D 3 ( αm 2 +( 1 5 +( 2 5 Therefore, sice 1 2 a 2 3, we have ( ( ( m+ 2 α m 1 m that is which leas to a the proof is complete. ( 1 α 5 m 1 m, 4 ( 5 + α, 5 5 2, ( αm 2 m 1 m. m 1 m, 14

16 7. A cocluig remark Give a fiite Abelia group G, the ivestigatio of the maximal possible legth of a zero-sumfree sequece S i G with large cross umber may also be of iterest. Cocerig this questio, we propose the followig geeral cojecture, which ca be see as a ual versio of Cojecture 1.2. Cojecture 7.1. Let G be a fiite Abelia group a G C ν1 C νs, with ν i > 1 for all i 1,s, be its logest possible ecompositio ito a irect prouct of cyclic groups. Give a zero-sumfree sequece S i G verifyig k(s k (G, oe always has the followig iequality: s S (ν i 1. It ca easily be see, by Theorem 1.1 (i, that Cojecture 7.1 hols true for fiite Abelia p-groups. Eve i the case of fiite cyclic groups which are ot p-groups, this problem is still wie ope. Yet, i this special case, the followig result supports the iea that a zero-sumfree sequece with large cross umber has to be a short sequece. Theorem 7.2. Let N be such that is ot a prime power, a let S be a zerosumfree sequece i C verifyig k(s k (C. The, oe has the followig iequality: S. 2 Proof. So as to prove this result, we will use the otio of iex of a sequece i a fiite cyclic group, which was itrouce implicitly i [16], Cojecture p.344, a more explicitly i [2]. Let g C with or(g =, a let S = (g 1,...,g l = ( 1 g,..., l g, where 1,..., l 0, 1, be a sequece i C. We efie: Sice, for every i 1,l, we have S g = gc( i, l = i. 1 or(g i, oe ca otice that S g k(s for all g C with or(g =. The, the iex of S, eote by iex(s, is efie i the followig fashio: iex(s = mi g C or(g = S g. Now, if is ot a prime power a S is a zero-sumfree sequece i C such that k(s k (C, oe obtais, by the very efiitio of the iex, the followig iequalities: iex(s k(s k (C > 1. 15

17 Therefore, usig a result of Savchev a Che (see Theorem 9 i [24], oe must have the followig iequality: S, 2 which completes the proof. I particular, Theorem 7.2 implies that Cojecture 7.1 hols true for all the cyclic groups of the form C 2p a, where p P a a N. Ackowlegmets I am grateful to my Ph.D. avisor Alai Plage for his help urig the preparatio of this paper. I woul like also to thak Alfre Geroliger a Wolfgag Schmi for useful remarks o a prelimiary versio of this work. Refereces [1] G. Bhowmik, I. Halupczok a J.C. Schlage-Puchta The structure of maximal zero-sum free sequeces II, submitte. [2] S. T. Chapma, M. Freeze a W.W. Smith Miimal zero sequeces a the strog Daveport costat, Discrete Math. 203 (1999, [3] F. Che a S. Savchev Miimal zero-sum sequeces of maximum legth i the group C 3 C 3k, Itegers 7 (2007, #A42. [4] W. Gao a A. Geroliger O log miimal zero sequeces i fiite abelia groups, Perio. Math. Hug. 38(3 (1999, [5] W. Gao a A. Geroliger O the orer of elemets i log miimal zero-sum sequeces, Perio. Math. Hug. 44(1 (2002, [6] W. Gao a A. Geroliger O zero-sum sequeces i Z/Z Z/Z, Itegers 3(2003, #A8. [7] W. Gao a A. Geroliger Zero-sum problems i fiite abelia groups : a survey, Expo. Math. 24 (2006, [8] W. Gao, A. Geroliger a D. Grykiewicz Iverse zero-sum problems III, submitte. [9] W. Gao, A. Geroliger a W. Schmi Iverse zero-sum problems, Acta Arith. 128 (2007, [10] A. Geroliger The cross umber of fiite abelia groups, J. Number Theory 48 (1994, [11] A. Geroliger a F. Halter-Koch No-uique factorizatios. Algebraic, combiatorial a aalytic theory, Pure a Applie Mathematics 278, Chapma & Hall/CRC (2006. [12] A. Geroliger a F. Halter-Koch No-uique factorizatios : a survey, Multiplicative ieal theory i commutative algebra, Spriger, New York (2006, [13] A. Geroliger a R. Scheier O miimal zero sequeces with large cross umber, Ars Combi. 46 (1997, [14] B. Girar A ew upper bou for the cross umber of fiite Abelia groups, Israel J. Math., to appear. [15] B. Girar Iverse zero-sum problems i fiite Abelia p-groups, submitte. [16] D. Kleitma a P. Lemke A aitio theorem o the itegers moulo, J. Number Theory 31 (1989,

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