Groups having many 2-generated subgroups in a given class
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If $\mathfrak{X}$ is a class of groups, denote by F$\mathfrak{X}$ the class of groups $G$ such that for every $x\in G$, there exists a normal subgroup of finite index $H(x)$ such that $\left\langle x,h\right\rangle \in\mathfrak{X}$ for every $h\in H(x)$. In this paper, we consider the class F$\mathfrak{X}$, when $\mathfrak{X}$ is the class of nilpotent-by-finite, finite-by-nilpotent and periodic-by-nilpotent groups. We will prove that for the above classes $\mathfrak{X}$ we have that a finitely generated hyper-(Abelian-by-finite) group in F$\mathfrak{X}$ belongs to $\mathfrak{X}$. As a consequence of these results, we prove that when the nilpotency class of the subgroups (or quotients) of the subgroups $\left\langle x,h\right\rangle$ are bounded by a given positive integer $k$, then the nilpotency class of the corresponding subgroup (or quotient) of $G$ is bounded by a positive integer $c$ depending only on $k$.
Abstract
If $\mathfrak{X}$ is a class of groups, denote by F$\mathfrak{X}$ the class of groups $G$ such that for every $x\in G$, there exists a normal subgroup of finite index $H(x)$ such that $\left\langle x,h\right\rangle \in\mathfrak{X}$ for every $h\in H(x)$. In this paper, we consider the class F$\mathfrak{X}$, when $\mathfrak{X}$ is the class of nilpotent-by-finite, finite-by-nilpotent and periodic-by-nilpotent groups. We will prove that for the above classes $\mathfrak{X}$ we have that a finitely generated hyper-(Abelian-by-finite) group in F$\mathfrak{X}$ belongs to $\mathfrak{X}$. As a consequence of these results, we prove that when the nilpotency class of the subgroups (or quotients) of the subgroups $\left\langle x,h\right\rangle$ are bounded by a given positive integer $k$, then the nilpotency class of the corresponding subgroup (or quotient) of $G$ is bounded by a positive integer $c$ depending only on $k$.
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