Geodesic
A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 4, pp. 975-982
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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.
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.
DOI : 10.21136/CMJ.2024.0271-23
Classification : 20E25, 20F19
Keywords: nilpotent; periodic; finite lower central depth; hyper-(Abelian-by-finite); minimal condition on normal subgroups 
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Gherbi, Fares; Trabelsi, Nadir. A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 4, pp. 975-982. doi: 10.21136/CMJ.2024.0271-23

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