Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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GROUPS HAVING MANY 2-GENERATED SUBGROUPS IN A GIVEN CLASS

  • Gherbi, Fares (Laboratory of Fundamental and Numerical Mathematics Department of Mathematics University Ferhat Abbas Setif 1) ;
  • Trabelsi, Nadir (Laboratory of Fundamental and Numerical Mathematics Department of Mathematics University Ferhat Abbas Setif 1)
  • Received : 2018.03.20
  • Accepted : 2018.06.21
  • Published : 2019.03.31
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Abstract

If 𝖃 is a class of groups, denote by F𝖃 the class of groups G such that for every $x{\in}G$, there exists a normal subgroup of finite index H(x) such that ${\langle}x,h{\rangle}{\in}$ 𝖃 for every $h{\in}H(x)$. 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 ${\langle}x,h{\rangle}$ 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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References

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