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.
Keywords : Nilpotent-by-finite groups, finite-by-nilpotent groups, periodic-by-nilpotent groups, Engel elements, hyper-(Abelian-by-finite) groups.
MSC numbers : 20F18, 20F45
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