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摘 要：如果在群CnCn中，每个含有2n-1个元素的极小零和序列中都包含一些阶数为n-1的元素，那么我们称正整数n具有性质B。在二维阿贝尔群的零和理论中，性质B是一个中心议题。关于性质B这一问题最早是由高维东教授和A.Geroldinger提出并进行研究[1-3]。之后，他们证明了如果n具有性质B[4-6]，当n大于等于6时，2n也具有性质；还证明了如果n∈{2，3，4，5，6，7}，n具有性质B。在文[7]中，我们证明了n=10时，n具有性质B。本文证明n=8时，n也具有性质B。
关键词：阿贝尔群；零和子列；性质B
DOI：10.15938/j.jhust.2017.06.021
中图分类号： O156.1
文献标志码： A
文章编号： 1007-2683（2017）06-0113-03
Abstract：We say a positive integer n has Property B if every minimal zerosum subsequence of 2n-1 elements in CnCn contains some elements n-1 times. Property B is a central topic in zerosum theory on abelian group G with rank two. Property B has been first formulated and investigated by professer W.D.Gao and A.Geroldinger in [1-3]. It has been proved that if n≥6 and if n has Property B， then 2n has Property B. It has been also proved that if n∈{2，3，4，5，6，7}， then n has property B[4-6]. In [7]， we proved that n=10 has Property B. In this paper， we will verify that n=8 has Property B.
Keywords：abelian group； zerosum subsequence； Property B
Similar to the proof of case 1， we can verify that there are at most two distinct elements in Tof case2 and case 3.
Theorem is true.
References：
[1] GAO W D， GEROLDINGER A. On Long Minimal Zero Sequences in Finite Abelian Groups[J]. Period Math. Hungar， 1999（38）：179-211.
[2] GAO W D， GEROLDINGER A. On Zerosum Sequences in Z/nZ Z/nZ[J]. Integers， 2003（3） （Paper A08）.
[3] GAO W D， ZHUANG J J. Sequences not Containing Long Zerosum Subsequences[J]. European J. Combin， 2006（27）： 777-787.
[4] GAO W D， GEROLDINGER A. Zerosum Probiems in Finite Abelian Groups： a survey[J]. Expo.Math， 2006（24）： 337-369.
[5] CHANG G J， CHEN S H， WANG G Q， et al. On the Number of Subsequences with a Given Sum in a Finite Abelian Group[J]. Electron. J. Combin， 2011（18）： 133-157.
[6] CHINTAMANI M N， MORIYA B K， GAO W D， et al. New Upper Bounds for the Davenport and for the ErdosGinzburgZiv Constants[J]. Arch. Math.， 2012（98）：133-142.
[7] ZANG H Y， LIU W H. A Property on Minimal Zerosum Subsequence inC10C10[J]. Advanced Materials Research ICEEIS2016981，2014：255-257.
[8] 韓冬春. ErdosGinzburgZiv Theorem for Finite Nilpotent Groups[J]. Arch. Math， 2015（104）： 325-332.
[9] GAO W D，LI Y L， YUAN P Z，et al. On the Structure of Long Zerosum Free Sequences and Nzerosum Free Sequences Over Finite Cyclic Groups[J]. Arch. Math， 2015（105）： 361-370.
[10]高维东，韩冬春，PENG J T， et al. On Zerosum Subsequences of Length k*exp（G）[J]. J. Combin. Theory Ser. 2014（A125）：240-253.
[11]ADHIKARIA S D，高维东，WANG G Q. ErdosGinzburgZiv Theorem for Finite Commutative Semigroups[J]. J. Pure Appl. Algebra， 2014（218）：1838-1844.
[12]高维东，路在平. The ErdsGinzburgZiv Theorem for Dihedral Groups[J]. J. Pure Appl. Algebra， 2008（212）： 311-319.
[13]高维东，侯庆虎，SCHMID W， et al. On Short Zerosum Subsequences II[J]. Integers，2007（7）：21-56.
[14]GAO W D， Alfred Geroldinger. On a Property of Minimal Zerosum Sequences and Restricted Sumsets[J]. Bull. London Math. Soc， 2005（37）：321-334.
[15]高维东，PENG J T，钟庆海. A Quantitative Aspect of Nonunique Factorizations： the Narkiewicz Constants III[J]. Acta Arith.， 2013（158）：271-285.
（编辑：温泽宇）
关键词：阿贝尔群；零和子列；性质B
DOI：10.15938/j.jhust.2017.06.021
中图分类号： O156.1
文献标志码： A
文章编号： 1007-2683（2017）06-0113-03
Abstract：We say a positive integer n has Property B if every minimal zerosum subsequence of 2n-1 elements in CnCn contains some elements n-1 times. Property B is a central topic in zerosum theory on abelian group G with rank two. Property B has been first formulated and investigated by professer W.D.Gao and A.Geroldinger in [1-3]. It has been proved that if n≥6 and if n has Property B， then 2n has Property B. It has been also proved that if n∈{2，3，4，5，6，7}， then n has property B[4-6]. In [7]， we proved that n=10 has Property B. In this paper， we will verify that n=8 has Property B.
Keywords：abelian group； zerosum subsequence； Property B
Similar to the proof of case 1， we can verify that there are at most two distinct elements in Tof case2 and case 3.
Theorem is true.
References：
[1] GAO W D， GEROLDINGER A. On Long Minimal Zero Sequences in Finite Abelian Groups[J]. Period Math. Hungar， 1999（38）：179-211.
[2] GAO W D， GEROLDINGER A. On Zerosum Sequences in Z/nZ Z/nZ[J]. Integers， 2003（3） （Paper A08）.
[3] GAO W D， ZHUANG J J. Sequences not Containing Long Zerosum Subsequences[J]. European J. Combin， 2006（27）： 777-787.
[4] GAO W D， GEROLDINGER A. Zerosum Probiems in Finite Abelian Groups： a survey[J]. Expo.Math， 2006（24）： 337-369.
[5] CHANG G J， CHEN S H， WANG G Q， et al. On the Number of Subsequences with a Given Sum in a Finite Abelian Group[J]. Electron. J. Combin， 2011（18）： 133-157.
[6] CHINTAMANI M N， MORIYA B K， GAO W D， et al. New Upper Bounds for the Davenport and for the ErdosGinzburgZiv Constants[J]. Arch. Math.， 2012（98）：133-142.
[7] ZANG H Y， LIU W H. A Property on Minimal Zerosum Subsequence inC10C10[J]. Advanced Materials Research ICEEIS2016981，2014：255-257.
[8] 韓冬春. ErdosGinzburgZiv Theorem for Finite Nilpotent Groups[J]. Arch. Math， 2015（104）： 325-332.
[9] GAO W D，LI Y L， YUAN P Z，et al. On the Structure of Long Zerosum Free Sequences and Nzerosum Free Sequences Over Finite Cyclic Groups[J]. Arch. Math， 2015（105）： 361-370.
[10]高维东，韩冬春，PENG J T， et al. On Zerosum Subsequences of Length k*exp（G）[J]. J. Combin. Theory Ser. 2014（A125）：240-253.
[11]ADHIKARIA S D，高维东，WANG G Q. ErdosGinzburgZiv Theorem for Finite Commutative Semigroups[J]. J. Pure Appl. Algebra， 2014（218）：1838-1844.
[12]高维东，路在平. The ErdsGinzburgZiv Theorem for Dihedral Groups[J]. J. Pure Appl. Algebra， 2008（212）： 311-319.
[13]高维东，侯庆虎，SCHMID W， et al. On Short Zerosum Subsequences II[J]. Integers，2007（7）：21-56.
[14]GAO W D， Alfred Geroldinger. On a Property of Minimal Zerosum Sequences and Restricted Sumsets[J]. Bull. London Math. Soc， 2005（37）：321-334.
[15]高维东，PENG J T，钟庆海. A Quantitative Aspect of Nonunique Factorizations： the Narkiewicz Constants III[J]. Acta Arith.， 2013（158）：271-285.
（编辑：温泽宇）