The cross number of minimal zero-sum sequences in finite abelian groups

Kim, Bumsoo

doi:10.1016/j.jnt.2015.04.012

Cited by 2 publications

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“…It is easy to see that K *( G )≤ K ( G ) and there is known no group for which inequality holds. For further progress on K ( G ), we refer to [ 5 , 15 , 16 , 18 ].…”

Section: Background On Transfer Krull Monoids and Sets Of Lengthsmentioning

confidence: 99%

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

Zhong

2018

Communications in Algebra

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Let H be a transfer Krull monoid over a finite abelian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit a∈H can be written as a product of irreducible elements, say , and the number of factors k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system ℒ(H) = {L(a)∣a∈H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system ℒ(H) is characteristic for the group G. Let H ′ be a further transfer Krull monoid over a finite abelian group G ′ and suppose that ℒ(H) = ℒ(H ′). We prove that, if with r≤n−3 or (r≥n−1≥2 and n is a prime power), then G and G ′ are isomorphic.

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“…It is easy to see that K *( G )≤ K ( G ) and there is known no group for which inequality holds. For further progress on K ( G ), we refer to [ 5 , 15 , 16 , 18 ].…”

Section: Background On Transfer Krull Monoids and Sets Of Lengthsmentioning

confidence: 99%

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

Zhong

2018

Communications in Algebra

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“…with ≥ p 5. For more information on ( ) η G and ( ) t G we refer to [1,[4][5][6][7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Tiny zero-sum sequences over some special groups

Wang

2020

Open Mathematics

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Let S={g}_{1}\cdot \ldots \cdot {g}_{n} be a sequence with elements {g}_{i} from an additive finite abelian group G. S is called a tiny zero-sum sequence if S is non-empty, {g}_{1}+\hspace{0.2em}\ldots \hspace{0.2em}+{g}_{n}=0 and k(S):= {\sum }_{i=1}^{n}\frac{1}{\text{ord}({g}_{i})}\le 1 . Let t(G) be the smallest integer t such that every sequence of t elements (repetition allowed) from G contains a tiny zero-sum sequence. In this article, we mainly focus on the explicit value of t(G) and compute this value for a new class of groups, namely ones of the form G={C}_{3}\oplus {C}_{3p} , where p is a prime number such that p\ge 5 .

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The cross number of minimal zero-sum sequences in finite abelian groups

Cited by 2 publications

References 10 publications

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

Tiny zero-sum sequences over some special groups

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