有限阿贝尔群中不含短零和子序列的序列研究

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"不含短零和子序列的序列的研究——曾祥能、袁平之的论文" 这篇论文"不含短零和子序列的序列"由曾祥能和袁平之撰写，探讨了有限阿贝尔群中一种特殊的序列类型。阿贝尔群是具有交换性质的群，这里的"序列"指的是在该群上的元素序列。文章的核心问题是研究那些不包含长度不大于特定值$h(S)$的零和子序列的序列$S$。零和子序列是指一个序列中的子序列，其元素相加结果为零。通常情况下，对于长度不小于阿贝尔群$G$大小的序列$S$，存在一个长度不超过$h(S)$的零和子序列。$h(S)$在这里被称为序列$S$的高度，它是一个与序列相关的量，通常表示找到一个零和子序列所需的最小步骤数。论文中提到的"逆问题"则是在已知序列$S$长度大于$|G|$且不含有长度不大于$h(S)$的零和子序列的情况下，去探究这样的序列$S$的具体结构。作者们在前人的工作基础上，即高、彭与王的成果（他们在假设$|∑(S)|<\min{|G|,2|S|−1}$时证明了序列$S$是"严格behaving"的），进一步推进了研究。他们放宽了这个条件，假设$|∑(S)|=2|S|−1<|G|$，并成功地给出了在这种情况下序列$S$的精确结构。关键词"零和问题"是数论领域的一个经典主题，关注的是如何在给定序列中找到零和子序列。"behaving序列"可能是指满足特定行为或性质的序列，而"高度"则与序列的结构和零和子序列的长度有关。"子和集"是指序列中所有可能子序列的和所构成的集合。这篇论文深入研究了阿贝尔群中特殊序列的结构，尤其是在限制零和子序列长度的情况下，为理解和构建这类序列提供了新的洞察。这一研究对于理解有限群的性质，以及在组合数论和编码理论等领域有潜在的应用价值。

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Naturally, we may ask what is the structure of S when 0 ∈



≤h(S)

(S), or more generally,

what can we say about such S. In [7], Gao et al. showed a result on the structure of such S.

To state the main theorem of [7], the authors introduced a deﬁnition which can be viewed as

the modiﬁcation of [8, Proposition 4] and [9, Deﬁnition 5.1.3].

Deﬁnition 2. ([7]) Let S ∈ F(G) be a sequence over an abelian group G. S is called strictly

g-behaving (strictly behaving for short) for some g ∈ G if S = (n

g)(n

g) · · · (n

g), where

|S| = r ∈ N, 1 = n

≤ · · · ≤ n

≤ ord(g) and n

≤



t−1

i=1

for all t ∈ [2, r].

Clearly, if S ∈ F(G) is strictly g-behaving, then



(S) = {g, 2g, . . . , ng} where n =

min{ord(g),



i=1

}. Also, if |S| ≥ 2, then h(S) ≥ v

(S) ≥ 2.

Theorem 3. ([7]) Let G be an abelian group and S ∈ F(G) a sequence such that ⟨supp(S)⟩ = G.

If 0 ∈



≤h(S)

(S), then either S is strictly behaving with respect to some g ∈ G or |



(S)| ≥

min{|G|, 2|S| − 1}.

As shown in [7], a lot of well-known results, including the ones in [8, 10, 11, 12, 13, 14],

can be viewed as special cases or corollaries of the theorem.

In the present paper, we take a step forward and give the structure of S such that the

equality in Theorem 3 holds.

Theorem 4. Let G be an abelian group and S ∈ F(G) a sequence with |S| = r and ⟨supp(S)⟩ =

G. Suppose that 0 ∈



≤h(S)

(S) and |



(S)| = 2r−1 ≤ |G|−1. Then S has one of the following

forms.

(I) S = (n

a)(n

a) · · · (n

a), where S is strictly a-behaving and n

+ · · · + n

= 2r − 1 ≤

ord(a) − 1.

(II) S = a

r−1

b, where ord(a) ≥ r and b ∈ {−(r − 2)a, . . . , −a, 0, a, 2a, . . . , (r − 1)a}.

(III) S = a

(a + e)

e, where ord(e) = 2, u ≥ v ≥ 0 and r = u + v + 1 ≤ ord(a + ⟨e⟩) in

G/⟨e⟩.

(IV) S = a

(a + e)

, where ord(e) = 2, u ≥ v ≥ 1 and r = u + v ≤ ord(a + ⟨e⟩) in G/⟨e⟩.

(V) S = a

r−2

bc, where ord(a) = r ≥ 3, b, c ∈ ⟨a⟩, b − c ∈ ⟨a⟩ \ {0} and b + c = a.

(VI) S = a

r−2

, where r ≥ 4, ord(a) = r, b ∈ ⟨a⟩ and 2b = 3a.

(VII) r = 3 and S = (−a)a(2a), where ord(a) ≥ 6.

(VIII) r = 4 and S = (−2a)

or S = (−2a)(−a)a(2a), where ord(a) ≥ 8.

(IX) r = 4 and S = (−a)a(−b)b, where ord(a) = 4, b ∈ ⟨a⟩ and 2b ∈ ⟨a⟩ \ {0}.

(X) r = 6 and S = a

(5a)

, where ord(a) = 12.

If S is a zero-sum free sequence or a subset of G \ {0}, then it is clear that 0 ∈



≤h(S)

(S).

Applying Theorems 3 and 4, we shall deduce the following two corollaries, which generalize

Theorem 1.2 of [14] and the main results of [15] respectively.

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有限阿贝尔群中不含短零和子序列的序列研究

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