Erdős-Ginzburg-Ziv theorem and Noether number for \(C_m \ltimes_\varphi C_{mn}\). (English) Zbl 1416.11014
In the present article, a multiplicative finite group \(G\) is considered. The main statements of this paper are proved for the cases when \(G\cong C_m \ltimes_{\varphi} C_{mn}\) and when \(G\) is a non-cyclic nilpotent group.
In this paper, the following main notions are called: the small Davenport constant denoted by \(\mathsf{d} (G)\), \(\mathsf{s}_L(G)\) (here \(L\) is a subset of \(\mathbb N\)), and \(\mathsf{e}(G)\), the Noether number \(\beta(G)\), as well as product-one, product-one free, and minimal product-one sequences. Also, it is noted (with definitions of these notions) that some classical product-one (zero-sum) invariants including \(\mathsf{D}(G)\), \(\mathsf{E}(G)\), \(\mathsf{s}(G)\), \(\eta(G)\), \(\mathsf{s}_{d\mathbb N}(G)\) (\(d\in\mathbb N\)) have received a lot of studies.
One can note the following main statements.
Theorem 1.2. Let \(G\) be a finite group, and \(m\) be any positive integer. If \(G\) has a normal subgroup \(N\) such that \(G/ N \cong C_m \ltimes_{\varphi} C_{m}\), then \[ |G|+ \mathsf{d}(G)\le\mathsf{E}(G)\le |G|+\frac{|G|}{m}+m-2. \] In particular, if \(G\cong C_m \ltimes_{\varphi} C_{mn}\), where \(m,n\) are positive integers, then \[ \mathsf{E}(G)=|G|+\mathsf{d} (G)=m^2n+mn+m-2. \] As a consequence, we have \[ \mathsf{d} (G)=mn+m-2. \]
Theorem 1.3. Let \(G\cong C_m \ltimes_{\varphi} C_{mn}\). We have \[ \mathsf{s}_{mn\mathbb N}(G)=m+2mn-2. \] If \(\mathsf{e}(G)=mn\), then we have \[ \eta(G)\le 2m+mn-2 \text{ and } \mathsf{s}(G)\le 2m+2mn-3. \]
Theorem 1.4. Let \(m,n\) be any positive integers, then \[ \beta\left( C_m \ltimes_{\varphi} C_{mn}\right)=mn+m-1. \]
In addition, the authors note that the result shows that \(\beta\left( C_m \ltimes_{\varphi} C_{mn}\right)=\mathsf{d}\left( C_m \ltimes_{\varphi} C_{mn}\right)+~1\).
Theorem 1.5. Let \(G\) be a non-cyclic nilpotent group and \(p\) the smallest prime divisor of \(|G|\), then \[ \beta(G)\le \frac{|G|}{p}+p-1 \] except if \(p=2\) and \(G\) is a dicyclic group, in which case \(\beta(G)=\frac{1}{2}|G|+2\).
Some auxiliary lemmas are proved. Based on obtained results, the authors propose the following conjecture.
Conjecture 6.1. For any positive integer \(n,m\) \[ \beta\left( C_n \ltimes_{\varphi} C_{m}\right)=n+m-1. \]
In this paper, the following main notions are called: the small Davenport constant denoted by \(\mathsf{d} (G)\), \(\mathsf{s}_L(G)\) (here \(L\) is a subset of \(\mathbb N\)), and \(\mathsf{e}(G)\), the Noether number \(\beta(G)\), as well as product-one, product-one free, and minimal product-one sequences. Also, it is noted (with definitions of these notions) that some classical product-one (zero-sum) invariants including \(\mathsf{D}(G)\), \(\mathsf{E}(G)\), \(\mathsf{s}(G)\), \(\eta(G)\), \(\mathsf{s}_{d\mathbb N}(G)\) (\(d\in\mathbb N\)) have received a lot of studies.
One can note the following main statements.
Theorem 1.2. Let \(G\) be a finite group, and \(m\) be any positive integer. If \(G\) has a normal subgroup \(N\) such that \(G/ N \cong C_m \ltimes_{\varphi} C_{m}\), then \[ |G|+ \mathsf{d}(G)\le\mathsf{E}(G)\le |G|+\frac{|G|}{m}+m-2. \] In particular, if \(G\cong C_m \ltimes_{\varphi} C_{mn}\), where \(m,n\) are positive integers, then \[ \mathsf{E}(G)=|G|+\mathsf{d} (G)=m^2n+mn+m-2. \] As a consequence, we have \[ \mathsf{d} (G)=mn+m-2. \]
Theorem 1.3. Let \(G\cong C_m \ltimes_{\varphi} C_{mn}\). We have \[ \mathsf{s}_{mn\mathbb N}(G)=m+2mn-2. \] If \(\mathsf{e}(G)=mn\), then we have \[ \eta(G)\le 2m+mn-2 \text{ and } \mathsf{s}(G)\le 2m+2mn-3. \]
Theorem 1.4. Let \(m,n\) be any positive integers, then \[ \beta\left( C_m \ltimes_{\varphi} C_{mn}\right)=mn+m-1. \]
In addition, the authors note that the result shows that \(\beta\left( C_m \ltimes_{\varphi} C_{mn}\right)=\mathsf{d}\left( C_m \ltimes_{\varphi} C_{mn}\right)+~1\).
Theorem 1.5. Let \(G\) be a non-cyclic nilpotent group and \(p\) the smallest prime divisor of \(|G|\), then \[ \beta(G)\le \frac{|G|}{p}+p-1 \] except if \(p=2\) and \(G\) is a dicyclic group, in which case \(\beta(G)=\frac{1}{2}|G|+2\).
Some auxiliary lemmas are proved. Based on obtained results, the authors propose the following conjecture.
Conjecture 6.1. For any positive integer \(n,m\) \[ \beta\left( C_n \ltimes_{\varphi} C_{m}\right)=n+m-1. \]
Reviewer: Symon Serbenyuk (Kyiv)
MSC:
| 11B13 | Additive bases, including sumsets |
| 11B50 | Sequences (mod \(m\)) |
| 11B75 | Other combinatorial number theory |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
Keywords:
zero-sum theory; Davenport constant; Erdős-Ginzburg-Ziv theorem; Noether number; product-one sequences; product-one free sequences; minimal product-one sequencesReferences:
| [1] | Bass, J., Improving the Erdős-Ginzburg-Ziv theorem for non-abelian groups, J. Number Theory, 126, 217-236 (2007) · Zbl 1179.20022 |
| [2] | Berkovich, Y., Groups of Prime Power Order, vol. I, de Gruyter Expositions in Mathematics (2008), de Gruyter: de Gruyter Berlin, New York · Zbl 1168.20001 |
| [3] | Cziszter, K., The Noether number of the non-abelian group of order \(3p\), Period. Math. Hungar., 68, 150-159 (2014) · Zbl 1349.13013 |
| [4] | Cziszter, K., On the Noether number of \(p\)-groups, J. Algebra Appl. (2018), in press |
| [5] | Cziszter, K.; Domokos, M., On the generalized Davenport constant and the Noether number, Cent. Eur. J. Math., 11, 1605-1615 (2013) · Zbl 1282.13012 |
| [6] | Cziszter, K.; Domokos, M., Groups with large Noether bound, Ann. Inst. Fourier (Grenoble), 64, 909-944 (2014) · Zbl 1314.13012 |
| [7] | Cziszter, K.; Domokos, M., The Noether number for the groups with a cyclic subgroup of index two, J. Algebra, 399, 546-560 (2014) · Zbl 1307.13009 |
| [8] | Cziszter, K.; Domokos, M., Lower bounds on the Noether number, Transform. Groups (2018) · Zbl 1434.13009 |
| [9] | Cziszter, K.; Domokos, M.; Geroldinger, A., The interplay of invariant theory with multiplicative ideal theory and with arithmetic combinatorics, (Multiplicative Ideal Theory and Factorization Theory. Multiplicative Ideal Theory and Factorization Theory, Springer Proceedings in Mathematics and Statistics, vol. 170 (2016), Springer), 43-95 · Zbl 1349.13014 |
| [10] | Cziszter, K.; Domokos, M.; Szőllősi, I., The Noether numbers and the Davenport constants of the groups of order less than 32, J. Algebra, 510, 513-541 (2018) · Zbl 1401.13018 |
| [11] | Domokos, M.; Hegedűs, P., Noether’s bound for polynomial invariants of finite groups, Arch. Math., 74, 161-167 (2000) · Zbl 0967.13004 |
| [12] | Erdős, P.; Ginzburg, A.; Ziv, A., Theorem in the additive number theory, Bull. Res. Counc. Isr., 10, 41-43 (1961) · Zbl 0063.00009 |
| [13] | Gao, W. D., A combinatorial problem on finite abelian groups, J. Number Theory, 58, 100-103 (1996) · Zbl 0892.11005 |
| [14] | Gao, W. D., An improvement of Erdős-Ginzburg-Ziv theorem, Acta Math. Sin., 39, 514-523 (1996) · Zbl 0862.20018 |
| [15] | Gao, W. D., On zero sum subsequences of restricted size II, Discrete Math., 271, 51-59 (2003) · Zbl 1089.11012 |
| [16] | Gao, W. D.; Geroldinger, A., Zero-sum problems in finite abelian groups: a survey, Expo. Math., 24, 337-369 (2006) · Zbl 1122.11013 |
| [17] | Gao, W. D.; Han, D. C.; Zhang, H. B., The EGZ-constant and short zero-sum sequences over finite abelian groups, J. Number Theory, 162, 601-613 (2016) · Zbl 1406.11019 |
| [18] | Gao, W. D.; Li, Y. L., The Erdős-Ginzburg-Ziv theorem for solvable groups, J. Pure Appl. Algebra, 214, 898-909 (2010) · Zbl 1195.20026 |
| [19] | Gao, W. D.; Lu, Z. P., The Erdős-Ginzburg-Ziv theorem for dihedral groups, J. Pure Appl. Algebra, 212, 311-319 (2008) · Zbl 1159.20326 |
| [20] | Geroldinger, A., Additive group theory and non-unique factorizations, (Combinatorial Number Theory and Additive Group Theory. Combinatorial Number Theory and Additive Group Theory, Adv. Courses Math. CRM Barcelona (2009), Birkhäuser Verlag: Birkhäuser Verlag Basel), 1-86 · Zbl 1221.20045 |
| [21] | Geroldinger, A.; Grynkiewicz, D. J., The large Davenport constant I: groups with a cyclic, index 2 subgroup, J. Pure Appl. Algebra, 217, 863-885 (2013) · Zbl 1276.20027 |
| [22] | Geroldinger, A.; Grynkiewicz, D. J.; Schmid, W. A., Zero-sum problems with congruence conditions, Acta Math. Hungar., 131, 323-345 (2013) · Zbl 1259.11020 |
| [23] | Geroldinger, A.; Halter-Koch, F., Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math., vol. 278 (2006), Chapman & Hall/CRC · Zbl 1113.11002 |
| [24] | Grynkiewicz, D. J., Structural Additive Theory, Developments in Mathematics, vol. 30 (2013), Springer: Springer Cham · Zbl 1368.11109 |
| [25] | Grynkiewicz, D. J., The large Davenport constant II: general upper bounds, J. Pure Appl. Algebra, 217, 2221-2246 (2013) · Zbl 1288.20029 |
| [26] | Halter-Koch, F., A generalization of Davenport’s constant and its arithmetical applications, Colloq. Math., 63, 203-210 (1992) · Zbl 0760.11031 |
| [27] | Han, D. C., The Erdős-Ginzburg-Ziv theorem for finite nilpotent groups, Arch. Math. (Basel), 104, 325-332 (2015) · Zbl 1347.11020 |
| [28] | Hegedűs, Pál; Maróti, Attila; Pyber, László, Finite groups with large Noether number are almost cyclic, Ann. Inst. Fourier (Grenoble) (2018), in press · Zbl 1425.13003 |
| [29] | Noether, E., Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., 77, 89-92 (1916) · JFM 45.0198.01 |
| [30] | Oh, J. S., On the algebraic and arithmetic structure of the monoid of product-one sequences, J. Commut. Algebra (2018), in press |
| [31] | Olson, J. E., On a combinatorial problem of Erdős, Ginzburg and Ziv, J. Number Theory, 8, 52-57 (1976) · Zbl 0333.05009 |
| [32] | Robinson, D., A Course in the Theory of Groups (1996), Springer |
| [33] | Schmid, B. J., Finite groups and invariant theory, (Topics in Invariant Theory. Topics in Invariant Theory, Lecture Notes in Mathematics, vol. 1478 (2003), Springer), 35-66 · Zbl 0770.20004 |
| [34] | Wehlau, D. L., The Noether number in invariant theory, C. R. Math. Acad. Sci. Soc. R. Can., 28, 39-62 (2006) · Zbl 1108.13008 |
| [35] | Yuster, T., Bounds for counter-example to addition theorem in solvable groups, Arch. Math. (Basel), 51, 223-231 (1988) · Zbl 0659.20020 |
| [36] | Yuster, T.; Peterson, B., A generalization of an addition theorem for solvable groups, Canad. J. Math., 36, 100-103 (1984) |
| [37] | Zhuang, J. J.; Gao, W. D., The Erdős-Ginzburg-Ziv theorem for dihedral groups of large prime index, European J. Combin., 26, 1053-1059 (2005) · Zbl 1077.20044 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
