Let G be an abelian group of order n. The critical number c(G) of G is the smallests such that the subset sums set Σ(S) covers all G for eachs ubset S⊂G\{0} of cardinality |S|≥s. It has been recently proved that, if p is the smallest prime dividing n and n/p is composite, then c(G)=|G|/p+p−2, thus establishing a conjecture of Diderrich. We characterize the critical sets with |S|=|G|/p+p−3 and Σ(S)=G, wh ere p≥3 is the smallest prime dividing n, n/p is composite and n≥7p2+3p. We also extend a result of Diderrichan d Mann by proving that, for n≥67, |S|≥n/3+2 and S=G imply Σ(S)=G. Sets of cardinality |S|≥ n+11 4 for which Σ(S)=G are also characterized when n≥183, the smallest prime p dividing n is odd and n/p is composite. Finally we obtain a necessary and sufficient condition for the equality Σ(G)=G to hold when |S|≥n/(p+2)+p, wh ere p≥5, n/p is composite and n≥15p2.

The following theorem is proved. Let 2≤k≤[In/4]+1, and let S be a sequence of 2n-k elements in Z,. Suppose that S does not contain any n-subsequence with 0-sum. Then, one can rearrange S to the type a,...a, b,...,b, c1,...,C2,-k-,-v, where u≥n-2k+3,

Let G be a finite abelian group (written additively), and let D (G) denote the Davenport's constant of G, i.e. the smallest integer d such that every sequence of d elements (repetition allowed) in G contains a nonempty zero-sum subsequence. Let S be a sequence of elements in G with |S|≥D (G). We say S is a normal sequence if S contains no zero-sum subsequence of length larger than |S|−D(G)+1. In this paper we obtain some results on the structure of normal sequences for arbitrary G. If G=Cn⊕Cn and nsatisfies some well-investigated property, we determine all normal sequences. Applying these results, we obtain correspondingly some results on the structure of the sequence S in G of length |S|=|G|+D(G)−2 and S contains no zero-sum subsequence of length |G|.

Let G be a finite abelian group. By Ol (G), we mean the smallest integert such that every subset A=G of cardinalityt contains a non-empty subset whose sum is zero. In this article, we shall prove that for all primes p>4.67×10*34, we have Ol (Zp⊕Zp)=p+Ol (Zp)-1 and hence we have Ol (Zp⊕Zp)≤p-1+「2p+5 logp「. This, in particular, proves that a conjecture of Erdo+s (stated below) is true for the group Zp⊕Zp for all primes p>4.67×10*34:

For any integer n≥3, by g (Zn⊕Zn) we denote the smallest positive integert such that every subset of cardinality t of the group Zn⊕Zn contains a subset of cardinality n whose sum is zero. Kemnitz (Extremalprobleme fur Gitterpunkte, Ph.D. Thesis, Technische Universitat Braunschweig, 1982) proved that g (Zp⊕Zp)=2p-1 for p=3,5,7, In this paper, as our main result, we prove that g (Zp⊕Zp)=2p-1 for all primes p≥67:

In this paper the following theorem is proved. Let G be a finite Abelian group of order n. Then, n+D (G)-1 is the least integer m with the property that for any sequence of m elements a1,..., am in G, 0 can be written in the form 0=a1+...+ain with 1≤i1<...<in≤m, where D (G) is the Davenport's constant on G, i.e., the least integer d with the property that for any sequence of d elements in G, there exists a nonempty subsequence that the sum of whose elements is 0.

Conjecture 0.1. Let S be a sequence of 3n&3 elements in Cn Cn. If S contains no nonempty zero-sum subsequence of length not exceeding n, then S consists of three distinct elements, each appearing n&1 times. Conjecture 0.2. Let S be a sequence of 4n&4 elements in Cn Cn. If S contains no zero-sum subsequence of length n, then S consists of four distinct elements, each appearing n-1 times. We show that both Conjecture 0.1 and Conjecture 0.2 are multiplicative, i.e., if Conjecture 0.1 (Conjecture 0.2) holds both for n=k and n=l then it holds also for n=kl.

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a1,..., ar that does not have 0 as a k-sum is attained at the sequence b1,..., br−k+1, 0,..., 0, where b1,..., br−k+1 is chosen to minimise the number of sums without 0 being a sum. Equivalently, to minimise the number of k-sums one should repeat some value k-1 times. This proves a conjecture of Bollobás and Leader, and extends results of Gao and of Bollobás and Leader.