\titleofpresentation{Centralizer of Engel elements in a group}

\presenter{Carmela Sica}  

This is joint work with G. Endimioni (Universit\'e de Provence,
Marseille (Fr)). 

Let $G$ be a group and $H$ a subgroup. Given some information about
the centralizer $C_G(H)$, what can we deduce about $G$? This type of
problem is well-known and has been studied by a number of people, see
for instance \cite{Sica1, Sica2, Sica4}. 
We are specially interested in the following result of Onishchuk and
Za\u\i tsev (\cite{Sica3}). 

\begin{proposition} 
Let $\mathfrak{X}$ denote the class of polycyclic groups, 
or
the class of \v{C}ernikov groups, 
or
the class of minimax groups, or the class of
groups of finite rank (in the sense of Pr\"{u}fer).  
Let $G$ be a nilpotent group, $H$ a finitely generated subgroup. Then
$C_G(H)$ belongs to  $\mathfrak{X}$ if and only if $G$ belongs to
$\mathfrak{X}$.  
\end{proposition}

From now on, the rank of a group is defined in the sense of
Pr\"{u}fer. Thus a group $G$ has finite rank if there exists an
integer $r$ such that every finitely generated subgroup 
of $G$ can be generated with at most $r$ elements; in this case, the
least integer $r$ satisfying this condition is the rank of $G$. 

An example given by the authors shows that their result can be false
if we consider a {\em locally} nilpotent group $G$.  
Let $A=\langle a_1\rangle \times \langle a_2\rangle \times\ldots
\times\langle a_n\rangle\times\ldots$ be an infinite elementary
abelian $p$-group and $B=\langle b\rangle$ be an infinite cyclic
group. Consider the semidirect product $G=A\rtimes B$, with
$a_1^b=a_1$ and $a_i^b=a_{n-1}a_n$ for $n>1$. Then $G$ is a metabelian
locally nilpotent group, with an element, $b$, whose centralizer is
polycyclic (thus minimax and of finite rank), and which is of infinite
rank (so it is neither polycyclic, nor minimax). Notice that this
example does not show that Proposition 1 fails for the class of
\v{C}hernikov groups. 

In the paper \cite{Sica1.1}, we aimed at obtaining similar results,
for instance when $G$ is soluble. In particular, we strengthen the
theorem of Onishchuk and  Za\u\i tsev when $\mathfrak{X}$ is the class
of polycyclic groups or the class of \v{C}hernikov groups. Obviously,
if we just assume that $G$ is soluble, we must formulate some extra
hypothesis about $H$. We can expect reasonably that $C_G(H)$ exercises
a strong influence on $G$ when the elements of $H$ are not very far
(in some sense) from the central elements. That leads to the notion of
Engel element, that can be considered as a generalization of the
notion of central element. We recall that in a group $G$, an element
$y$ is said to be a {\em left Engel} element if for all $x$ in $G$,
there exists an integer $m\geq 0$ such that $[x,_my]=1$ (as usual,
$[x,_my]$ is inductively defined  by $[x,_0y]=x$ and
$[x,_my]=[x,_{m-1}y]^{-1}y^{-1}[x,_{m-1}y]y$ when $m>0$). If $m$ can
be chosen independently of $x$, one says that $y$ is a  {\em bounded}
left Engel element. We shall write $L(G)$ for the set of left Engel
elements of $G$, and $\overline{L}(G)$ for the set of bounded left
Engel elements. 

Our first result generalizes Proposition 1 when  $\mathfrak{X}$ is the
class of polycyclic groups. 

\begin{theorem} Let $G$ be a soluble group. Let $H$ be a finitely
generated subgroup contained in $\overline{L}(G)$ and suppose that
$C_G(H)$ is polycyclic. Then $G$ is polycyclic. 
\end{theorem}
 
The example in \cite{Sica3} shows also that Theorem 2 fails if we
replace $\overline{L}(G)$ by $L(G)$ in its statement. 
As we noticed above, Onishchuk and Za\u\i tsev did not say if
Proposition 1 is still true for locally nilpotent groups in which the
centralizer is a \v{C}hernikov group. Our next theorem gives a
positive answer. 
Before stating this result, recall also that a {\em radical group} is
a group having an ascending series with locally nilpotent
quotients. The class of radical groups is very large, containing for
instance all soluble groups and all locally nilpotent groups. 

\begin{theorem} Let $G$ be a radical group. Let $H$ be a finitely
generated subgroup contained in  $L(G)$ and suppose that $C_G(H)$ is a
\v{C}hernikov group. Then $G$ is a soluble \v{C}hernikov group.  
\end{theorem}

Finally, we have proved that Proposition 1 holds for a {\em periodic}
locally nilpotent group $G$ when $\mathfrak{X}$ is the class of groups
of finite rank (with this choice of $\mathfrak{X}$, we have seen that  
Proposition 1 fails for an arbitrary locally nilpotent group). 

\begin{theorem} Let $G$ be a periodic locally nilpotent group. Let $H$
be a finitely generated subgroup such that $C_G(H)$ has finite
rank. Then $G$ has finite rank.  
\end{theorem}
%

\begin{ourbiblio}{99} 

\bibitem{Sica1} G. Endimioni, Groupes dont le centralisateur d'un sous-groupe est d'ordre born\'{e}.
{\em Math. Ann.} {\bf 291} (1991), 475--479. 

\bibitem{Sica1.1} G. Endimioni and C. Sica, Centralizer of Engel elements in a group.

\bibitem{Sica2} E.I. Khukhro and V.D. Mazurov, Finite groups with an automorphism of prime order whose centralizer has small rank. {\em J. Algebra} {\bf 301} (2006), 474--492. 

\bibitem{Sica3} V.A Onishchuk and D.I. Za\u\i tsev, On locally nilpotent groups with a centralizer 
that satisfies a finiteness condition. {\em Ukrainian Math. J.} {\bf 43} (1991), 1018--1021;

\bibitem{Sica4} V.P. \v{S}unkov, On periodic groups with an almost regular involution. {\em Algebra and Logic} {\bf 11} (1972), 260--272. 


\end{ourbiblio}

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

\email{csica@unisa.it}

