Groups having many 2-generated subgroups in a given class

Bull. Korean Math. Soc. Published online 2019 Mar 11

Fares GHERBI, and Nadir TRABELSI
Department of Mathematics, Faculty of Sciences, University Ferhat Abbas Setif 1

Abstract : If X is a class of groups, denote by FX the class of groups G such that for every x∈G, there exists a normal subgroup of finite index H(x) such that <x,h>∈X for every h∈H(x). In this paper, we consider the class FX, when X is the class of nilpotent-by-finite, finite-by-nilpotent and periodic-by-nilpotent groups. We will prove that for the above classes X we have that a finitely generated hyper-(Abelian-by-finite) group in FX belongs to X. As a consequence of these results, we prove that when the nilpotency class of the subgroups (or quotients) of the subgroups <x,h> 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.