Article
| Title: | A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent (English) |
| Author: | Gherbi, Fares |
| Author: | Trabelsi, Nadir |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 4 |
| Year: | 2024 |
| Pages: | 975-982 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $\mathfrak {M}$ be the class of groups satisfying the minimal condition on normal subgroups and let $\Omega $ be the class of groups of finite lower central depth, that is groups $G$ such that $\gamma _{i}(G)=\gamma _{i+1}(G)$ for some positive integer $i$. The main result states that if $G$ is a finitely generated hyper-(Abelian-by-finite) group such that for every $x\in G$, there exists a normal subgroup $H_{x}$ of finite index in $G$ satisfying $\langle x,x^{h}\rangle \in \mathfrak {M}\Omega $ for every $h\in H_{x}$, then $G$ is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated hyper-(Abelian-by-finite) group $G$ such that for every $x\in G$, there exists a normal subgroup $H_{x}$ of finite index in $G$ satisfying $\langle x,x^{h}\rangle \in \mathfrak {T}\Omega $ for every $h\in H_{x}$, is periodic-by-nilpotent; where $\mathfrak {T}$ stands for the class of periodic groups. (English) |
| Keyword: | nilpotent |
| Keyword: | periodic |
| Keyword: | finite lower central depth |
| Keyword: | hyper-(Abelian-by-finite) |
| Keyword: | minimal condition on normal subgroups |
| MSC: | 20E25 |
| MSC: | 20F19 |
| DOI: | 10.21136/CMJ.2024.0271-23 |
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| Date available: | 2024-12-15T06:33:36Z |
| Last updated: | 2024-12-16 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152685 |
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| Reference: | [1] Brookes, C. J. B.: Engel elements of soluble groups.Bull. Lond. Math. Soc. 18 (1986), 7-10. Zbl 0556.20028, MR 841360, 10.1112/blms/18.1.7 |
| Reference: | [2] Falco, M. De, Giovanni, F. De, Musella, C., Trabelsi, N.: A nilpotency-like condition for infinite groups.J. Aust. Math. Soc. 105 (2018), 24-33. Zbl 1498.20080, MR 3820253, 10.1017/S1446788717000416 |
| Reference: | [3] Gherbi, F., Trabelsi, N.: Groups having many 2-generated subgroups in a given class.Bull. Korean Math. Soc. 56 (2019), 365-371. Zbl 1512.20108, MR 3936471, 10.4134/BKMS.b180247 |
| Reference: | [4] Gherbi, F., Trabelsi, N.: Groups with restrictions on some subgroups generated by two conjugates.Adv. Group Theory Appl. 12 (2021), 35-45. Zbl 1495.20040, MR 4353905, 10.32037/agta-2021-010 |
| Reference: | [5] Gherbi, F., Trabelsi, N.: Properties of Abelian-by-cyclic shared by soluble finitely generated groups.Turk. J. Math. 46 (2022), 912-918. Zbl 1509.20054, MR 4406735, 10.55730/1300-0098.3131 |
| Reference: | [6] Golod, E. S.: Some problems of Burnside type.Twelve Papers on Algebra, Algebraic Geometry and Topology American Mathematical Society Translations: Series 2, Vol. 84. AMS, Providence (1969), 83-88. Zbl 0206.32402, MR 0238880, 10.1090/trans2/084 |
| Reference: | [7] Hammoudi, L.: Burnside and Kurosh problems.Int. J. Algebra Comput. 14 (2004), 197-211. Zbl 1087.16013, MR 2058320, 10.1142/S0218196704001694 |
| Reference: | [8] Lennox, J. C.: Finitely generated soluble groups in which all subgroups have finite lower central depth.Bull. Lond. Math. Soc. 7 (1975), 273-278. Zbl 0314.20029, MR 382448, 10.1112/blms/7.3.273 |
| Reference: | [9] Lennox, J. C.: Lower central depth in finitely generated soluble-by-finite groups.Glasg. Math. J. 19 (1978), 153-154. Zbl 0394.20027, MR 486159, 10.1017/S0017089500003554 |
| Reference: | [10] Robinson, D. J. S.: Finiteness Conditions and Generalized Soluble Groups. Part 1.Ergebnisse der Mathematik und ihrer Grenzgebiete 62. Springer, New York (1972). Zbl 0243.20032, MR 0332989, 10.1007/978-3-662-07241-7 |
| Reference: | [11] Robinson, D. J. S.: Finiteness Conditions and Generalized Soluble Groups. Part 2.Ergebnisse der Mathematik und ihrer Grenzgebiete 63. Springer, New York (1972). Zbl 0243.20033, MR 0332990, 10.1007/978-3-662-11747-7 |
| Reference: | [12] Robinson, D. J. S.: A Course in The Theory of Groups.Graduate Texts in Mathematics 80. Springer, New York (1996). Zbl 0836.20001, MR 1357169, 10.1007/978-1-4419-8594-1 |
| Reference: | [13] Segal, D.: A residual property of finitely generated abelian-by-nilpotent groups.J. Algebra 32 (1974), 389-399. Zbl 0293.20029, MR 419612, 10.1016/0021-8693(74)90147-1 |
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