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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 $k$.
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참고문헌 (10)
참고문헌 신청
C. J. B. Brookes / 1986 / Engel elements of soluble groups / Bull. London Math. Soc 18(1):7~10
M. DE FALCO / 2018 / A NILPOTENCY-LIKE CONDITION FOR INFINITE GROUPS / Journal of the Australian Mathematical Society 105(1):24~33
S. Franciosi / 2000 / Groups with many polycyclic-by-nilpotent subgroups / Ricerche Mat 48(2):361~378
E. S. Golod / 1969 / Some problems of Burnside type / Amer. Math. Soc. Transl. Ser. 2 84:83~88
L. Hammoudi / 2004 / Burnside and Kurosh problems / Internat. J. Algebra Comput 14(2):197~211
M. DE FALCO / 2018 / A NILPOTENCY-LIKE CONDITION FOR INFINITE GROUPS / Journal of the Australian Mathematical Society 105(1):24~33
S. Franciosi / 2000 / Groups with many polycyclic-by-nilpotent subgroups / Ricerche Mat 48(2):361~378
E. S. Golod / 1969 / Some problems of Burnside type / Amer. Math. Soc. Transl. Ser. 2 84:83~88
L. Hammoudi / 2004 / Burnside and Kurosh problems / Internat. J. Algebra Comput 14(2):197~211
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