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Keywords:
nilpotent; periodic; finite lower central depth; hyper-(Abelian-by-finite); minimal condition on normal subgroups
nilpotent; periodic; finite lower central depth; hyper-(Abelian-by-finite); minimal condition on normal subgroups
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.
References:
[1] Brookes, C. J. B.: Engel elements of soluble groups. Bull. Lond. Math. Soc. 18 (1986), 7-10. DOI 10.1112/blms/18.1.7 | MR 841360 | Zbl 0556.20028
[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. DOI 10.1017/S1446788717000416 | MR 3820253 | Zbl 1498.20080
[3] Gherbi, F., Trabelsi, N.: Groups having many 2-generated subgroups in a given class. Bull. Korean Math. Soc. 56 (2019), 365-371. DOI 10.4134/BKMS.b180247 | MR 3936471 | Zbl 1512.20108
[4] Gherbi, F., Trabelsi, N.: Groups with restrictions on some subgroups generated by two conjugates. Adv. Group Theory Appl. 12 (2021), 35-45. DOI 10.32037/agta-2021-010 | MR 4353905 | Zbl 1495.20040
[5] Gherbi, F., Trabelsi, N.: Properties of Abelian-by-cyclic shared by soluble finitely generated groups. Turk. J. Math. 46 (2022), 912-918. DOI 10.55730/1300-0098.3131 | MR 4406735 | Zbl 1509.20054
[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. DOI 10.1090/trans2/084 | MR 0238880 | Zbl 0206.32402
[7] Hammoudi, L.: Burnside and Kurosh problems. Int. J. Algebra Comput. 14 (2004), 197-211. DOI 10.1142/S0218196704001694 | MR 2058320 | Zbl 1087.16013
[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. DOI 10.1112/blms/7.3.273 | MR 382448 | Zbl 0314.20029
[9] Lennox, J. C.: Lower central depth in finitely generated soluble-by-finite groups. Glasg. Math. J. 19 (1978), 153-154. DOI 10.1017/S0017089500003554 | MR 486159 | Zbl 0394.20027
[10] Robinson, D. J. S.: Finiteness Conditions and Generalized Soluble Groups. Part 1. Ergebnisse der Mathematik und ihrer Grenzgebiete 62. Springer, New York (1972). DOI 10.1007/978-3-662-07241-7 | MR 0332989 | Zbl 0243.20032
[11] Robinson, D. J. S.: Finiteness Conditions and Generalized Soluble Groups. Part 2. Ergebnisse der Mathematik und ihrer Grenzgebiete 63. Springer, New York (1972). DOI 10.1007/978-3-662-11747-7 | MR 0332990 | Zbl 0243.20033
[12] Robinson, D. J. S.: A Course in The Theory of Groups. Graduate Texts in Mathematics 80. Springer, New York (1996). DOI 10.1007/978-1-4419-8594-1 | MR 1357169 | Zbl 0836.20001
[13] Segal, D.: A residual property of finitely generated abelian-by-nilpotent groups. J. Algebra 32 (1974), 389-399. DOI 10.1016/0021-8693(74)90147-1 | MR 419612 | Zbl 0293.20029
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