INVERSE PROBLEMS OF SUBSET SUMS OF ZERO-SUM FREE SET WITH SIX ELEMENTS IN FINITE ABELIAN GROUPS
Keywords:
abelian groups, inverse problems, subset sums, zero-sum free setDOI:
https://doi.org/10.17654/0972555522036Abstract
Let $S$ be a subset of an additive finite abelian group, and $\Sigma(S)$ denote the sumset of $S$, which is defined as the set of nonempty subset sums of $S$. In this paper, we prove that if $|S|=6$ and $0 \notin \Sigma(S)$, then $|\Sigma(S)|=20$ if and only if the subgroup generated by $S$ is a cyclic group of 21 elements.
Received: October 2, 2022
Accepted: November 9, 2022
References
G. Bhowmik, I. Halupczok and J. C. Schlage-Puchta, Zero-sum free sequences with small sum-set, Math. Comp. 80 (2011), 2253-2258.
R. B. Eggleton and P. Erdos, Two combinatorial problems in group theory, Scientific paper 117, Dept. of Math., Stat. and Comp. Sci., U. of Calgary, 1971.
É. Balandraud, B. Girard, S. Griffiths and Y. Hamidoune, Subset sums in abelian groups, European J. Combin. 34 (2013), 1269-1286.
R. B. Eggleton and P. Erdos, Two combinatorial problems in group theory, Acta Arith. 21 (1972), 111-116.
L. Chen, G. Li, Y. Wen, H. Yang, J. Liu and J. Peng, On the inverse problems of subsums of zero-sum free subsets, JP Journal of Algebra, Number Theory and Applications 41(1) (2019), 19-33.
M. Freeze, W. Gao and A. Geroldinger, The critical number of finite abelian groups, J. Number Theory 129 (2009), 2766-2777.
W. Gao and A. Geroldinger, On the structure of zerofree sequences, Combinatorica 18 (1998), 519-527.
W. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey, Egpo. Math. 24 (2006), 337-369.
W. Gao, M. Huang, W. Hui, Y. Li, C. Liu and J. Peng, Sums of sets of abelian group elements, J. Number Theory 208 (2020), 208-229.
W. Gao and I. Leader, Sums and k-sums in abelian groups of order k, J. Number Theory 120 (2006), 26-32.
W. Gao, Y. Li, J. Peng and F. Sun, Subsums of a zero-sum free subset of an abelian group, Electron J. Combin. 15 (2008), Research Paper 116, 36 pp.
A. Geroldinger, Additive group theory and non-unique factorizations, Combinatorial Number Theory and Additive Group Theory, Adv. Course Math. CRM Barcelona, 2009, pp. 1-86.
A. Geroldinger and F. Halter-Koch, Non-unique factorizations, Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. (Boca Raton), 278, 2006.
A. Geroldinger and Y. O. Hamidoune, Zero-sumfree sequences in cyclic groups and some arithmetical application, J. Théor. Nombres Bordeaux 14 (2002), 221-239.
H. Guan, G. Zeng and P. Yuan, Description of invariant F(5) of a zero-sum free sequence, Acta. Sci. Natur. Univ. Sunyatseni 49 (2010), 1-4 (in Chinese).
J. Li and D. Wan, Counting subset sums of finite abelian groups, J. Combin. Theory A 119 (2012), 170-182.
M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer Berlin, 1996.
J. E. Olson, An addition theorem mod p, J. Combin. Theory 5 (1968), 45-52.
J. E. Olson, Sums of sets of group elements, Acta Arith. 28 (1975), 147-156.
J. Peng and W. Hui, On the structure of zero-sum free set with minimum subset sums in abelian groups, Ars Combin. 146 (2019), 63-74.
P. Yuan and G. Zeng, On zero-sum free subsets of length 7, Electron. J. Combin. 17 (2010), Research Paper 104, 13 pp.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Pushpa Publishing House, Prayagraj, India
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
