The Erdős–Ginzburg–Ziv theorem for finite solvable groups

Gao, Weidong; Li, Yuanlin

doi:10.1016/j.jpaa.2009.08.003

Cited by 18 publications

(19 citation statements)

References 12 publications

(8 reference statements)

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“…The Erdős-Ginzburg-Ziv constant s(G) is the smallest integer ∈ N such that every sequence S over G of length |S| ≥ has a zero-sum subsequence of length n. The famous Erdős-Ginzburg-Ziv theorem states that s(G) = 2|G| − 1 if G is cyclic [8]. The Erdős-Ginzburg-Ziv constant has found far reaching generalizations (more information can be found in the surveys [14,21]; for recent progress see [1,2,4,7,9,16,22,31]). …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the structure of n-zero-sum free sequences over cyclic groups of order n

Peng

Gong

Sun

et al. 2014

Int. J. Number Theory

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Let G be a finite cyclic group of order n. The Erdős-Ginzburg-Ziv theorem states that each sequence of length 2n−1 over G has a zero-sum subsequence of length n. A sequence without a zero-sum subsequence of length n is called n-zero-sum free. Savchev and Chen characterized all the n-zero-sum free sequences of length n + k − 1 over G, where n 2 + 1 ≤ k < n. In the present paper, we determine all the n-zero-sum free sequences of length n + n 2 − 1 over G.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the structure of n-zero-sum free sequences over cyclic groups of order n

Peng

Gong

Sun

et al. 2014

Int. J. Number Theory

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“…Lemma 2.5. ( [18]) Let G be a non-cyclic finite solvable group of order n. Then every sequence over G of length kn + 3 4 n − 1 contains a product-one subsequence of length kn.…”

Section: Preliminariesmentioning

confidence: 99%

Erdős–Ginzburg–Ziv theorem and Noether number for C⋉C

Han

Zhang

2019

Journal of Number Theory

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Let G be a multiplicative finite group and S = a 1 · . . . · a k a sequence over G. We call S a product-one sequence if 1 = k i=1 a τ (i) holds for some permutation τ of {1, . . . , k}. The small Davenport constant d(G) is the maximal length of a product-one free sequence over G. For a subsethave received a lot of studies. The Noether number β(G) which is closely related to zero-sum theory is defined to be the maximal degree bound for the generators of the algebra of polynomial invariants. Let G ∼ = Cm ⋉ϕ Cmn, in this paper, we prove that E(G) = d(G) + |G| = m 2 n + m + mn − 2 and β(G) = d(G) + 1 = m + mn − 1. We also prove that s mnN (G) = m + 2mn − 2 and provide the upper bounds of η(G), s(G). Moreover, if G is a non-cyclic nilpotent group and p is the smallest prime divisor of |G|, we prove that β(G) ≤

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“…Suppose that i = 3 and N is a non-trivial proper normal subgroup of G. Consider all possible value of the pair |N |, |G/N | ∈ (2, 36), (4,18), (6,12), (8,9), (9,8), (12,6), (18,4), (36,2) .…”

Section: Arithmetic Of the Monoid Of Product-one Sequencesmentioning

confidence: 99%

“…For i = 4, let N be a non-trivial proper normal subgroup of G and consider all possible value of the pair |N |, |G/N | ∈ (2, 72), (3,48), (4,36), (6,24), (8,18), (9,16), (12,12), (16,9), (18,8), (24,6), (36,4), (48, 3), (72, 2) .…”

Section: Arithmetic Of the Monoid Of Product-one Sequencesmentioning

confidence: 99%

“…The focus on the present paper is on nonabelian finite groups. Sequences over general (not necessarily abelian) finite groups have been studied in combinatorics since the work of Olson ([30] for an upper bound on the small Davenport constant) and there has been renewed interest ( [2,13,12,27,15,26,4]), partly motivated by connections to invariant theory ( [9,6,8,7,10]).…”

Section: Introductionmentioning

confidence: 99%

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On the algebraic and arithmetic structure of the monoid of product-one sequences II

2018

Period Math Hung

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Let G be a finite group and G ′ its commutator subgroup. By a sequence over G, we mean a finite unordered sequence of terms from G, where repetition is allowed, and we say that it is a productone sequence if its terms can be ordered such that their product equals the identity element of G. The monoid B(G) of all product-one sequences over G is a finitely generated C-monoid whence it has a finite commutative class semigroup. It is well-known that the class semigroup is a group if and only if G is abelian (equivalently, B(G) is Krull). In the present paper we show that the class semigroup is Clifford (i.e., a union of groups) if and only if |G ′ | ≤ 2 if and only if B(G) is seminormal, and we study sets of lengths in B(G).

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The Erdős–Ginzburg–Ziv theorem for finite solvable groups

Cited by 18 publications

References 12 publications

On the structure of n-zero-sum free sequences over cyclic groups of order n

On the structure of n-zero-sum free sequences over cyclic groups of order n

Erdős–Ginzburg–Ziv theorem and Noether number for C⋉C

On the algebraic and arithmetic structure of the monoid of product-one sequences II

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