
%  When you are satisfied with your abstract, please delete everything
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\titleofpresentation{On minimal non torsion-by-nilpotent and non (locally finite)-by-nilpotent groups}

\presenter{Nadir Trabelsi}  %  Name just one speaker, please; collaborators
			%  can be credited in the body of the abstract.

%  [Body of abstract goes here.]
Denote by $\mathcal{N}$ (respectively, $\mathcal{N}%
_{c}$, $\mathcal{F}$) the class of nilpotent (respectively, nilpotent of
class at most $c$, finite) groups and let $\mathcal{X}$ stands for the class of torsion groups or the class of
locally finite groups. If $\Omega $ is a class of groups, then a group is said to be minimal non-$%
\Omega $ if it is not an $\Omega $-group, while all its proper subgroups
belong to $\Omega $. Many results have been obtained on minimal non-$\Omega $%
, for various classes of groups $\Omega $. In particular, in \cite{trabelsi1}
it is proved that if $G$ is an infinite finitely
generated minimal non-$\mathcal{N}$ group, then $G/Frat(G)$ is an infinite
simple group, where $Frat(G)$ stands for the Frattini subgroup of $G$. Also
in \cite{trabelsi3} it is proved that if $G$ is a finitely generated minimal non-$%
\mathcal{FN}$ group, then $G$ is a perfect group which has no proper
subgroup of finite index and $G/Frat(G)$ is an infinite simple group. In
this note we obtain a similar result on minimal non-$\mathcal{XN}$ groups
and we prove that there are no minimal non-$\mathcal{XN}$ groups which are
not finitely generated. More precisely, we prove the following result:

\begin{theorem}
If $G$\ is a minimal non-$\mathcal{XN}$\ group, then $G$\ is a finitely
generated perfect group which has no proper subgroup of finite index and
$G/Frat(G)$\ is an infinite simple group.
\end{theorem}

Using a well known result of Za\"{i}cev \cite{trabelsi4} which asserts that an infinite
nilpotent group whose proper subgroups are in the class $\mathcal{N}_{c}$ is
itself in $\mathcal{N}_{c}$, we can see that Theorem 1 has the following
consequence:

\begin{corollary}
If $G$\ is a minimal non-$\mathcal{XN}_{c}$\ group, then $G$\ is a finitely
generated perfect group which has no proper subgroup of finite index and
$G/Frat(G)$\ is an infinite simple group.
\end{corollary}

Among minimal non-$\mathcal{XN}$ groups and minimal non-$\mathcal{XN}_{c}$
groups we can cite the example constructed by Ol'shanskii \cite{trabelsi2}
of an infinite simple torsion-free finitely generated group whose proper
subgroups are cyclic.

\begin{ourbiblio}{99}  %  Delete this environment if you
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\bibitem{trabelsi1} M. F. Newman and J. Wiegold, Groups with many nilpotent
subgroups, Arch. Math. \textbf{15} (1964), 241-250.

\bibitem{trabelsi2} A. Y. Ol'shanskii, An infinite simple
torsion-free Noetherian group, Izv. Akad. Nauk SSSR Ser. Mat. \textbf{43}
(1979), 1328-1393.

\bibitem{trabelsi3} M. Xu, Groups whose proper subgroups are finite-by-nilpotent,
Arch. Math. \textbf{66} (1996), 353-359.

\bibitem{trabelsi4}  D. I. Zaicev, Stably nilpotent groups, Mat. Zametki \textbf{2}
(1967), 337-346.


\end{ourbiblio}

\presenteraddress{University of Setif - Algeria}  % The name of a university is fine.

\email{nadir_trabelsi@yahoo.fr}

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