
\titleofpresentation{Locally Graded Quotients\\of Locally Graded Groups}
\presenter{Maria Tota}  


The presentation will be about a joint work with Akbar Rhemtulla
(University of Alberta, Canada), on locally graded groups.

A group
$G$ is said to be locally graded if every nontrivial, finitely
generated subgroup of $G$ has a nontrivial finite image. The class
$\mathcal{L}$ of locally graded groups is quite wide (locally soluble
groups and locally residually finite groups belong to $\mathcal{L}$)
and it frequently appears in literature (\cite{mtota1}, \cite{mtota2},
\cite{mtota3}, \cite{mtota4}, and the papers cited in these) mainly in
the study of groups that do not have infinite, finitely generated
simple subgroups.

It can be easily seen that a subgroup of an
$\mathcal{L}$-group is an $\mathcal{L}$-group but the class
$\mathcal{L}$ is not closed under forming quotients, for free groups
are locally graded thus every group can occur as a quotient of an
$\mathcal{L}$-group.

Hence, naturally arises the problem of looking
for $\mathcal{L}^{QS}$, the $\{$$Q$-$S$$\}$-closed interior of
$\mathcal{L}$, that is the largest subclass of $\mathcal{L}$ which is
closed with respect to forming sections.

If we denote by
$\mathcal{L}_1$ the class of all groups in which every finitely
generated simple section is finite, it turns out that
$\mathcal{L}_1=\mathcal{L}^{QS}$.

Moreover, we give conditions under
which a quotient 
of a locally graded group is locally graded.

In fact, we generalize
the result proved in \cite{mtota3}, stating that if $G$ belongs to
$\mathcal{L}$ and $N$ is a locally nilpotent, normal subgroup of $G$,
then the factor group $G/N$ still belongs to $\mathcal{L}$.

Then, we
focus on linear groups which are in particular locally graded and we
get more precise conditions for a quotient of a linear group to be
locally graded. 

\begin{ourbiblio}{99}  

\bibitem{mtota1} N.S. Chernikov, D.Ya. Trebenko,
\textit{Quotient groups of locally graded groups and of groups of some Kuros-Chernikov classes.}
Ukrainian Math. J. \textbf{50} (1998), no.11, 1765-1773 (1999).


\bibitem{mtota2} Y. Kim and A.H. Rhemtulla,
\textit{On locally graded groups.}
Groups-Korea '94 (Pusan), de Gruyter, Berlin (1995), 189-197.

\bibitem{mtota3} P. Longobardi, M. Maj, H. Smith,
\textit{A Note on Locally Graded Groups.}
Rend. Sem. Mat. Univ. Padova, \textbf{94} (1995), 275-277.

\bibitem{mtota4} H. Smith,
\textit{On homomorphic images of locally graded groups.}
Rend. Sem. Mat. Univ. Padova, \textbf{91} (1994), 53-60.


\end{ourbiblio}

\presenteraddress{Universit\`a di Salerno (Italy)}  

\email{mtota@unisa.it}

\web{www.dmi.unisa.it/people/tota/www/}  

