On Subsequence Sums Of Zero-Sum Free Sequences In Abelian Groups

This thesis focuses on the subsequence sums of zero-sum free sequences in finite abelian groups.It is a crossover study of combinatorial mathematics and number theory,which can be summed up in combinatorial number theory that a very active research field in the world.In recent years,the study of subsequence sums has attracted many scholars including R.B.Eggleton,P.Erdos,J.E.Olson,I.Leader,A.Pixton,GAO Weidong,YUAN Pingzhi.This article focuses on a special kind of sequences--sets.Let G be an abelian group and S(?)G be a subset.Let ?(S)denote the set of group elements which can be expressed as a sum of a nonempty subset of S.This article shows the research from two different points of view of the direct problem and the inverse problem.The so-called direct problem is to determine the number of elements of subset sums and the inverse problem is to determine the structure of S,which has the smallest number of elements of subset sums.Suppose S is zero-sum free,i.e.0(?)?(S).This paper consists of three parts.First of all,the inverse problem of a zero-sum free subset of 6 elements.It was conjectured by R.B.Eggleton and P.Erdos in 1972 and proved by GAO Weidong et al.in 2008 that if |S}=6,then |?z(S)|?19.In this paper,the structure of the zero-sum free set S when |S|=6 and |?(S)|=19 has been determined.Besides,the inverse problem of a zero-sum free subset of 7 elements.It was proved by YUAN Pingzhi et al.in 2010 that if |S|=7,then |?(S)|?24.In this paper,|?(S)|=24 if and only if(S)is a cyclic group with order 25 has been proved.Finally,the number of subset sums of a zero-sum free subset of 8 elements.Combining with the former two parts,it has been proved that if |S|=8,then |?(S)|?30.It is noteworthy that at present the exact lower bound of |?(S)| is still unable to determine.