A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent

Fares Gherbi; Nadir Trabelsi

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 4, page 975-982
  • ISSN: 0011-4642

Abstract

top
Let 𝔐 be the class of groups satisfying the minimal condition on normal subgroups and let Ω be the class of groups of finite lower central depth, that is groups G such that γ i ( G ) = γ 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 ∈ G , there exists a normal subgroup H x of finite index in G satisfying 〈 x , x h 〉 ∈ 𝔐 Ω for every h ∈ 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 ∈ G , there exists a normal subgroup H x of finite index in G satisfying 〈 x , x h 〉 ∈ 𝔗 Ω for every h ∈ H x , is periodic-by-nilpotent; where 𝔗 stands for the class of periodic groups.

How to cite

top

Gherbi, Fares, and Trabelsi, Nadir. "A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent." Czechoslovak Mathematical Journal 74.4 (2024): 975-982. <http://eudml.org/doc/299644>.

@article{Gherbi2024,
abstract = {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.},
author = {Gherbi, Fares, Trabelsi, Nadir},
journal = {Czechoslovak Mathematical Journal},
keywords = {nilpotent; periodic; finite lower central depth; hyper-(Abelian-by-finite); minimal condition on normal subgroups},
language = {eng},
number = {4},
pages = {975-982},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent},
url = {http://eudml.org/doc/299644},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Gherbi, Fares
AU - Trabelsi, Nadir
TI - A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 4
SP - 975
EP - 982
AB - 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.
LA - eng
KW - nilpotent; periodic; finite lower central depth; hyper-(Abelian-by-finite); minimal condition on normal subgroups
UR - http://eudml.org/doc/299644
ER -

References

top
  1. Brookes, C. J. B., 10.1112/blms/18.1.7, Bull. Lond. Math. Soc. 18 (1986), 7-10. (1986) Zbl0556.20028MR841360DOI10.1112/blms/18.1.7
  2. Falco, M. De, Giovanni, F. De, Musella, C., Trabelsi, N., 10.1017/S1446788717000416, J. Aust. Math. Soc. 105 (2018), 24-33. (2018) Zbl1498.20080MR3820253DOI10.1017/S1446788717000416
  3. Gherbi, F., Trabelsi, N., 10.4134/BKMS.b180247, Bull. Korean Math. Soc. 56 (2019), 365-371. (2019) Zbl1512.20108MR3936471DOI10.4134/BKMS.b180247
  4. Gherbi, F., Trabelsi, N., 10.32037/agta-2021-010, Adv. Group Theory Appl. 12 (2021), 35-45. (2021) Zbl1495.20040MR4353905DOI10.32037/agta-2021-010
  5. Gherbi, F., Trabelsi, N., 10.55730/1300-0098.3131, Turk. J. Math. 46 (2022), 912-918. (2022) Zbl1509.20054MR4406735DOI10.55730/1300-0098.3131
  6. Golod, E. S., 10.1090/trans2/084, Twelve Papers on Algebra, Algebraic Geometry and Topology American Mathematical Society Translations: Series 2, Vol. 84. AMS, Providence (1969), 83-88. (1969) Zbl0206.32402MR0238880DOI10.1090/trans2/084
  7. Hammoudi, L., 10.1142/S0218196704001694, Int. J. Algebra Comput. 14 (2004), 197-211. (2004) Zbl1087.16013MR2058320DOI10.1142/S0218196704001694
  8. Lennox, J. C., 10.1112/blms/7.3.273, Bull. Lond. Math. Soc. 7 (1975), 273-278. (1975) Zbl0314.20029MR382448DOI10.1112/blms/7.3.273
  9. Lennox, J. C., 10.1017/S0017089500003554, Glasg. Math. J. 19 (1978), 153-154. (1978) Zbl0394.20027MR486159DOI10.1017/S0017089500003554
  10. Robinson, D. J. S., 10.1007/978-3-662-07241-7, Ergebnisse der Mathematik und ihrer Grenzgebiete 62. Springer, New York (1972). (1972) Zbl0243.20032MR0332989DOI10.1007/978-3-662-07241-7
  11. Robinson, D. J. S., 10.1007/978-3-662-11747-7, Ergebnisse der Mathematik und ihrer Grenzgebiete 63. Springer, New York (1972). (1972) Zbl0243.20033MR0332990DOI10.1007/978-3-662-11747-7
  12. Robinson, D. J. S., 10.1007/978-1-4419-8594-1, Graduate Texts in Mathematics 80. Springer, New York (1996). (1996) Zbl0836.20001MR1357169DOI10.1007/978-1-4419-8594-1
  13. Segal, D., 10.1016/0021-8693(74)90147-1, J. Algebra 32 (1974), 389-399. (1974) Zbl0293.20029MR419612DOI10.1016/0021-8693(74)90147-1

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.