On Finite Extension and Conditions on Infinite Subsets of Finitely Generated FC and FN_k-groups

Year 2018, Issue: 23, 22 - 30, 01.06.2018

Mourad Chelgham Mohamed Kerada Lemnouar Noui

Abstract

Let k>0 an integer. F, τ,
N, N_k, and A denote, respectively, the classes of
finite, torsion, nilpotent, nilpotent of class at most k, group in which every
two generator subgroup is in N_k and abelian groups. The main results
of this paper is, firstly, to prove that in the class of finitely generated
FN-group, the property FC is closed under finite extension. Secondly, we prove
that a finitely generated τN-group in the class ((τN_k)τ,∞) ( respectively
((τN_k)τ,∞)^∗)
is a τ-group (respectively τN_c
for certain integer c=c(k) ) and deduce that a finitely generated FN-group in
the class ((FN_k)F,∞) (respectively ((FN_k)F,∞)^∗)
is -group (respectively FN_c
for certain integer c=c(k)). Thirdly we prove that a finitely generated
NF-group in the class ((FN_k)F,∞) ( respectively ((FN_k)F,∞)^∗)
is F-group (respectively N_cF
for certain integer c=c(k)). Finally and particularly, we deduce that a
finitely generated FN-group in the class ((FA)F,∞) (respectively ((FC)F,∞)^∗,
((FN₂)F,∞)^∗)
is in the class FA (respectively FN₂,
FN₃⁽²⁾).

Keywords

FC-group, (FC)F-group, (τNk)τ-group, (FNk)F-group, ((FNk)F, ∞)-group, ∞)∗-group, finitely generated group

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