A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS


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1 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 A CHARACTERIZATION OF MINIMAL ZEROSEQUENCES OF INDEX ONE IN FINITE CYCLIC GROUPS Scott T. Chapma 1 Triity Uiversity, Departmet of Mathematics, Oe Triity Place, Sa Atoio, TX , USA William W. Smith The Uiversity of North Carolia at Chapel Hill, Departmet of Mathematics, Phillips Hall, Chapel Hill, NC , USA Received: 4/27/05, Revised: 10/26/05, Accepted: 11/16/05, Published: 11/29/05 Abstract Let G = Z where is a positive iteger. A fiite sequece S = {g 1,..., g k } of ot ecessarily distict elemets from G for which k g i = 0 is called a zerosequece. If a zerosequece S cotais o proper subzerosequece, the it is called a miimal zerosequece. The otio of the idex of a miimal zerosequece (see Defiitio 1) i Z has bee recetly addressed i the mathematical literature. I this ote, we offer a characterizatio of miimal zerosequeces i Z with idex 1. Let G be a additive abelia group ad S = {g 1,..., g k } a fiite sequece of ot ecessarily distict elemets from G. Deote by S = k the umber of elemets i S (or the legth of S) ad let supp(s) = {g g G with g = g i for some i} be the support of S. Various properties of the sequece S have bee cosidered over the last several years i the mathematical literature. Some of these properties are amog the followig. 1. S is zerofree if i I g i 0 for ay oempty subset I {1, 2,..., k}. 2. S is a zerosequece if k g i = A zerosequece S is a miimal zerosequece (or MZS) if for every oempty I {1, 2,..., k}, the sequece {g i } i I is zerofree. 1 Part of this work was completed while the first author was o a Academic Leave grated by the Triity Uiversity Faculty Developmet Committee.
2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A A zerosequece S which is ot a MZS is a almost miimal zerosequece (or AMZS) if for every oempty I {1, 2,..., k} where the sequece {g i } i I is a zerosequece, the {g i } i I is a miimal zerosequece. I this article, we will cosider a property of miimal zerosequeces i fiite cyclic groups which was itroduced i the literature i [2] ad cosequetly cosidered i greater detail i [4] ad [7]. Some otatio will be ecessary before givig a formal statemet describig this property. Sice the orderig of the elemets i a sequece S does ot matter, we will view sequeces as elemets of F(G), the free abelia mooid o G. Hece, we write S = g G g g where oly fiitely may of the g are ot zero. Our goal is to offer a characterizatio of idex 1 miimal zerosequeces i Z. This will be doe i terms of almost miimal zerosequeces (see [3, Chapter 5] for more iformatio o AMZSs). We will fid the laguage of block mooids useful for expressig ad applyig some of our argumets. For a fiite abelia group G, let B(G) represet the set of elemets i F(G) which are zerosequeces. Further, let U(G) be the subset of B(G) cosistig of the miimal zerosequeces of G. If S 1 = g G gmg ad S 2 = g G gsg are i B(G), the B(G) ca be cosidered as a commutative cacellative mooid uder the operatio S 1 S 2 = g G g mg+sg ad is commoly called a block mooid (more iformatio o block mooids ca be foud i [6]). The irreducible elemets of B(G) are merely the elemets of U(G) ad the empty block (i.e., S = ) acts as the idetity of B(G). A iterpretatio of a almost miimal zerosequece i terms of block mooids ca be stated as follows: B B(G) is a almost miimal zerosequece if ad oly if B = B 1 B t with each B i i U(G) implies that t = 2. Defiitio 1. Let G be a abelia group. (1) Let g G be a ozero elemet with ord(g) = > 1. For a sequece S = ( 1 g) ( l g), where l N 0 ad 1,..., l [1, ], we defie S g = l to be the gorm of S. If S =, the set S g = 0. (2) Let S be a zerosum sequece for which supp(s) G is a otrivial fiite cyclic group. The we call the idex of S. idex(s) = mi{ S g g G with supp(s) = g } N 0
3 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 3 Notice that the idex of a sequece S depeds oly o S ad ot the choice of the cyclic group G which cotais supp(s). Theorem 2 of [2] idicates that as icreases, there exist miimal zerosequeces of Z of arbitrarily high idex. The papers [7] ad [4] have both show that for a fixed value of, log miimal zerosequeces must have idex 1. I particular, [4, Sectio 2] shows for 10 that a miimal zerosequece S i Z with S > 2 must have idex 1. 3 Whe restrictig our attetio to cyclic groups, the gorm of a zerosequece ca be used to draw some helpful coclusios. We determie some basic properties of the gorm i the ext propositio. Propositio 2. Let G be a abelia group, g G a ozero elemet ad S, T B( g ). (1) g : B( g ) N 0 is a mooid homomorphism (i.e., S T g = S g + T g ). (2) S g = 0 if ad oly if S =. (3) 0 g = 1. (4) If S g = 1, the S is a MZS. (5) If S g = 2, the S is a AMZS. Proof. The proofs of (1)(3) are clear. For (4), if S = S 1 S 2 with S 1 ad S 2 i B( g ), the 1 = S g = S 1 g + S 2 g 2, a cotradictio. For (5), if S is either a MZS or a AMZS, the S = S 1 S 2 S 3 for S 1, S 2 ad S 3 i B( g ). The argumet ow follows as i (4). We ote that idex oe MZSs satisfy several iterestig properties. Two of these properties follow. Recall that if S = g G gg is a MZS i Z, the the cross umber of S is defied as k(s) = g g G where ord(g) represets the order of g i G (more iformatio ord(g) o the cross umber ca be foud i [1]). For S B(G) cosider these properties. (P1) S S is a AMZS i Z. (P2) k(s) 1. It follows directly from Propositio 2 that S = g G gg a MZS i Z with S g = 1 satisfies (P1). That S g = 1 implies k(s) 1 ca be see as follows. Suppose S = ( 1 g) ( l g) is writte as i Defiitio 1 with = ord(g). The k(s) = l Hece we have the followig. 1 ord( i g) = l 1 gcd ( i,) k i = S g = 1. Propositio 3. If S is a MZS of Z with idex(s) = 1, the S satisfies properties (P1) ad (P2).
4 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 4 Example 4. Properties (P1) ad (P2) do ot characterize MZSs of idex 1. Notice that all of the idex 2 MZSs i [2] do ot satisfy (P1) (see i particular the proof of [2, Theorem 2]). A slight modificatio of the costructio used i [2] yields the followig example. Let G = Z 23 ad set S = It is a routie calculatio to check the 22 possible values of S g ad determie that idex(s) = 2. Sice k(s) 1, S satisfies (P2). For cosiderig property (P1), ote that S 1 = 2 ad so S S 1 = 4. To establish that S S is a AMZS, oe eeds oly observe that if it were ot, the S S = A B C for some zero sequeces A, B, ad C. It follows that this has to be doe (with the proper choice of g) so that A g = B g = 1 ad C g = 2. The key the to observig such a decompositio is impossible is to ote that is the oly subsequece of S S that sums to 23. While (P1) ad (P2) do ot offer the characterizatio of idex 1 MZSs we desire, a relatively simple coditio ivolvig the AMZS s which cotai a MZS S does provide a characterizatio. Theorem 5. Let G be a abelia group ad S a miimal zerosequece over G such that supp(s) geerates a cyclic group H of order 2. The the followig statemets are equivalet: (a) There exists some AMZS A F(H) of legth A = S + where S divides A i B(G). (b) There exists some g H such that g S is a AMZS. (c) idex(s) = 1. Proof. (a) (b) Let A = ST be a AMZS of legth S + for some T F(H). The T is a miimal zerosum sequece of legth. Thus, for example by [5, Lemma 13], there exists some g H such that T = g. (b) (c) Let g H ad A = g S a AMZS. The there are m 1,..., m l [1, 1] with m 1... m l such that S = l (m ig). We assert that S g = 1. Assume to the cotrary that S g = m m l = k with k 2. Sice S is a miimal zerosum sequece, there exist u, v [1, l 1] such that ad We set (k 2) < m m u < (k 1) < m m u + m u+1 m u m v < < m u m v + m v+1. r = (k 1) (m m u ),
5 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A27 5 ad we defie s = (m u m v ) u v l N 1 = g r (m i g), N 2 = g s (m i g) ad N 3 = g (r+s) (m i g). i=u+1 i=v+1 By costructio, N 1, N 2 ad N 3 are zerosum sequeces with A = N 1 N 2 N 3, a cotradictio to the fact that A is a AMZS. (c) (a) Let g H such that S g = 1. We set A = g S, ad sice A g = 2, it follows that A is a AMZS. Ackowledgemet The authors would like to thak the referee for may helpful suggestios. Refereces [1] S. T. Chapma, O the Daveport costat, the cross umber, ad their applicatio i factorizatio theory, Zerodimesioal commutative rigs (Koxville, TN, 1994), Marcel Dekker, New York, 1997, [2] S. T. Chapma, M. Freeze, ad W. W. Smith, Miimal zerosequeces ad the strog Daveport costat, Discrete Math. 203(1999), [3] M. Freeze, Legths of Factorizatios i Dedekid domais, Ph. D. Dissertatio, The Uiversity of North Carolia at Chapel Hill, [4] W. D. Gao, Zero sums i fiite cyclic groups, Itegers, (electroic) 0(2000) A12, 7 pp. [5] A. Geroldiger, Systeme vo Lägemege, Abh. Math. Sem. Uiv. Hamburg 60(1990), [6] A. Geroldiger ad F. HalterKoch, Nouique factorizatios i block semigroups ad their arithmetical applicatios, Math. Slovaca 42(1992), [7] V. Poomareko, Miimal zero sequeces of fiite Abelia groups, Itegers, (electroic) 4(2004) A24, 6 pp.
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