On Modified Erdős-Ginzburg-Ziv constants of finite abelian groups

Yuting Hu; Jiangtao Peng; Mingrui Wang; Yuting Hu; Jiangtao Peng; Mingrui Wang

doi:10.3934/math.2023339

AIMS Mathematics

2023, Volume 8, Issue 3: 6697-6704. doi: 10.3934/math.2023339

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Research article

On Modified Erdős-Ginzburg-Ziv constants of finite abelian groups

College of Science, Civil Aviation University of China, Tianjin, China 300300

Received: 01 September 2022 Revised: 22 December 2022 Accepted: 30 December 2022 Published: 09 January 2023
MSC : 11B13, 11P70, 05D10

Let $ G $ be a finite abelian group with exponent $ \exp(G) $ and $ S $ be a sequence with elements of $ G $. We say $ S $ is a zero-sum sequence if the sum of the elements in $ S $ is the zero element of $ G $. For a positive integer $ t $, let $ \mathtt{s}_{t\exp(G)}(G) $ (respectively, $ \mathtt{s}'_{t\exp(G)}(G) $) denote the smallest integer $ \ell $ such that every sequence (respectively, zero-sum sequence) $ S $ over $ G $ with $ |S|\geq \ell $ contains a zero-sum subsequence of length $ t\exp(G) $. The invariant $ \mathtt{s}_{t\exp(G)}(G) $ (respectively, $ \mathtt{s}'_{t\exp(G)}(G) $) is called the Generalized Erdős-Ginzburg-Ziv constant (respectively, Modified Erdős-Ginzburg-Ziv constant) of $ G $. In this paper, we discuss the relationship between Generalized Erdős-Ginzburg-Ziv constant and Modified Erdős-Ginzburg-Ziv constant, and determine $ \mathtt{s}'_{t\exp(G)}(G) $ for some finite abelian groups.
- zero-sum sequence,
- Erdős-Ginzburg-Ziv Theorem,
- Generalized Erdős-Ginzburg-Ziv constant,
- Modified Erdős-Ginzburg-Ziv constant
Citation: Yuting Hu, Jiangtao Peng, Mingrui Wang. On Modified Erdős-Ginzburg-Ziv constants of finite abelian groups[J]. AIMS Mathematics, 2023, 8(3): 6697-6704. doi: 10.3934/math.2023339

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Abstract

Let $ G $ be a finite abelian group with exponent $ \exp(G) $ and $ S $ be a sequence with elements of $ G $. We say $ S $ is a zero-sum sequence if the sum of the elements in $ S $ is the zero element of $ G $. For a positive integer $ t $, let $ \mathtt{s}_{t\exp(G)}(G) $ (respectively, $ \mathtt{s}'_{t\exp(G)}(G) $) denote the smallest integer $ \ell $ such that every sequence (respectively, zero-sum sequence) $ S $ over $ G $ with $ |S|\geq \ell $ contains a zero-sum subsequence of length $ t\exp(G) $. The invariant $ \mathtt{s}_{t\exp(G)}(G) $ (respectively, $ \mathtt{s}'_{t\exp(G)}(G) $) is called the Generalized Erdős-Ginzburg-Ziv constant (respectively, Modified Erdős-Ginzburg-Ziv constant) of $ G $. In this paper, we discuss the relationship between Generalized Erdős-Ginzburg-Ziv constant and Modified Erdős-Ginzburg-Ziv constant, and determine $ \mathtt{s}'_{t\exp(G)}(G) $ for some finite abelian groups.

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