
%  When you are satisfied with your abstract, please delete everything
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\titleofpresentation{Rational endomorphisms of a nilpotent group}

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

%  [Body of abstract goes here.]
Let $G$ be a group. A function $f:G\rightarrow G$ is said to be \emph{%
rational} if there exist elements $a_{1},\dots ,a_{r},a_{r+1}\in G$ and
integers $h_{1},\dots ,h_{r}\in \mathbb{Z}$ such that 
\[
f(x)=a_{1}x^{h_{1}}a_{2}x^{h_{2}}\cdots a_{r}x^{h_{r}}a_{r+1}\;\;\text{for
all}\;\;x\in G.
\]
If the exponents $h_{1},\dots ,h_{r}$ can be chosen to be positive, we say
that $f$ is \emph{polynomial}; it is the case when for example the group $G$
is of finite exponent. The set $F(G)$ of all rational functions of $G$ forms
a monoid with the usual composition of functions, this monoid has two
important submonoids, $\mathcal{F}(G)$ the set of rational functions $f$
such that $f(1)=1$, and $End_{r}(G)$ the set of rational functions which are
endomorphisms. It is obvious that $End_{r}(G)\subseteq \mathcal{F}(G)$. We
denote by $F^{*}(G),\mathcal{F}^{*}(G)$, and $End_{r}^{*}(G)$, the group of
invertible elements of the corresponding monoids. Note that a bijective
rational function of a group $G$ is not necessarily in $F^{*}(G)$ (e.g.,
consider $G=\left( 0,\infty \right) $ with multiplication and $f:x\mapsto
x^{3}$). On the other hand, if $G$ is finite, then $F^{*}(G)=Sym(G)\cap F(G)$%
, $End_{r}^{*}(G)=Sym(G)\cap End_{r}(G),$ where $Sym(G)$ is the set of all
bijections on $G$.

Following \cite{Daoud2}, we denote by $J(G)$ the set of integers $m$ such that
there exist $t$ integers $m_{1},\dots ,m_{t}$ ($t\geq 1$) and $t$ elements $%
c_{1},\dots ,c_{t}$ of $G$ such that $m=m_{1}+\cdots +m_{t}$ and $%
c_{1}x^{m_{1}}\cdots c_{t}x^{m_{t}}=1$ for all $x\in G$. The set $J(G)$ is
an ideal of the ring of integers $\mathbb{Z}$ (see \cite[p. 431]{Daoud2}).
Let $m\geq 0$ be the integer such that $J(G)=(m)$ and $f\in F(G)$, where $%
f(x)=a_{1}x^{h_{1}}\cdots a_{r}x^{h_{r}}a_{r+1}$. Put $h=h_{1}+\cdots +h_{r}$
and $h_{0}\in \{0,\dots ,m-1\}$ is the unique integer such that $h_{0}-h\in
J(G)$. If $h_{0}=h+qm$ for some integer $q$, then $f(x)=f(x)(c_{1}x^{m_{1}}%
\cdots c_{t}x^{m_{t}})^{q}$. Thus we may assume that $h\in \{0,\dots ,m-1\}$
and we call $\bar{h}$ the \emph{degree} of $f$. If $f(1)=1$ (i.e., $f\in \mathcal{F}(G))$, then 
\[
f(x)=x^{h}\prod_{i=1}^{r}[x^{h_{i}},b_{i}][x^{h_{i}},b_{i},x^{h-(h_{1}+%
\cdots +h_{i})}],
\]
where $b_{i}=(a_{1}\cdots a_{i})^{-1}$ for all $i\in \{1,\dots ,r\}$. 

Endimioni \cite{Daoud2} has proved that if $G$ is nilpotent, $F^{*}(G)$ is a
nilpotent-by-(finite abelian) group. In this paper we prove that

\begin{theorem}
If $G$ is a group, we denote by $\mathcal{F}_{1}(G)$ the set of elements o $%
\mathcal{F}(G)$ of degree $\bar{1}$. Then, if $G$ is nilpotent of class $%
c\geq 2$, we have 

(i) \  $\mathcal{F}_{1}(G)$ is a normal subgroup of $\mathcal{F}^{*}(G)$,
and $\frac{\mathcal{F}^{*}(G)}{\mathcal{F}_{1}(G)}$ is isomorphic to the
group of invertible elements of the ring $\frac{\mathbb{Z}}{J(G)}$.

(ii) \  $\mathcal{F}_{1}(G)$ is nilpotent of class $c-1$.
\end{theorem}

Schweigert \cite[Satz 3.5, p.37]{Daoud3} showed that if $G$ is a finite group,
then $G$ is nilpotent if and only if $End_{r}^{*}(G)$ is nilpotent. Corsi
Tani and Rinaldi Bonafede \cite[Theorem 3.5, p. 288]{Daoud1} improved
Schweigert's result by proving that if $G$ is a finite group, then it is
nilpotent of class $c\geq 2$ if and only if $End_{r}^{*}(G)$ is nilpotent of
class $c-1$. Note that if $G$ is abelian, then $End_{r}^{*}(G)=\mathcal{F}%
^{*}(G)$ coincides with the group of universal power automorphisms which is
contained in the centre of $Aut(G)$, the automorphisms group of $G$. We
generalize \cite[Theorem 3.5, p. 288]{Daoud1} for infinite groups, in fact we
prove

\begin{theorem}
Let $G$ be a group and $c\geq 3$ be an integer. Then $G$ is
nilpotent of class $c$ if and only if $End_{r}^{*}(G)$ is nilpotent of class 
$c-1$. Moreover, $End_{r}^{*}(G)$ is normal in $Aut(G)$.
\end{theorem}

It was proved in \cite[Theorem 5.2, p. 291]{Daoud1} that if $G$ is a finite
group and $r\geq 2$ is a given integer, then the sum of any $r$ inner
automorphisms of $G$ is an automorphism of $G$ (i.e., the map $x\mapsto
x^{a_{1}}\cdots x^{a_{r}}$ is an automorphisms of $G$ for any $r$ elements $%
a_{1},\dots ,a_{r}\in G$) if and only if $G$ is 2-Engel with the property
that $\binom{r}{2}\equiv 0\;\text{mod}G^{\prime }$ and $\gcd (r,\text{exp}%
(G))=1$. Here we generalize this result to any group. 

Let $h$ be an integer. We say that a group $G$ satisfies $\mathcal{I}(h)$ $(%
\mathcal{E}(h))$ if the map $x\longmapsto \left( x^{a_{1}}\right)
^{h_{1}}\ldots \left( x^{a_{r}}\right) ^{h_{r}}$ is an invertible rational
endomorphism (epimorphism) of $G$ for all $a_{1},a_{2},\dots ,a_{r}\in G$
and $h_{1},h_{2},\dots ,h_{r}\in \mathbb{Z}$ such that $h=h_{1}+\cdots
+h_{r}.$ 

\begin{theorem}
Let $G$ be a group and $h$ an integer different of $-1,0,1$.
Then $G$ satisfies $\mathcal{I}(h)$ if and only if $G$ is 2-Engel with
conditions:

1)\ exp($G$) is finite and $\gcd (h,$exp$(G))=1.$


2) $\text{exp}(G^{\prime })$ divides $h(h-1)/2$. 

If $G$ satisfies $\mathcal{E}(h)$, then $G$ is a 2-Engel group with the only
property that $\text{exp}(G^{\prime })$ is finite and divides $h(h-1)/2$.
\end{theorem}


\begin{ourbiblio}{99}  %  Delete this environment if you
		       %  aren't giving a bibliography.

\bibitem{Daoud1} G. Corsi Tani and M.F. Rinaldi Bonafede, \textsl{Polynomial
automorphisms in nilpotent finite groups}, Boll. Un. Mat. Ital. A (6) 
\textbf{5} (1986), no. 2, 285-292.

\bibitem{Daoud2} G. Endimioni, \textsl{Applications rationnelles d'un groupe
nilpotent}, C. R. Acad. Sci. Paris S\'{e}r. I Math. \textbf{314} (1992), no.
6, 431-434.

\bibitem{Daoud3} D. Schweigert, \textsl{Polynomautomorphismen auf endlichen
gruppen}, Arch. Math. (Basel) \textbf{29} (1977), no. 1, 34-38.

\end{ourbiblio}

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

\email{boun_daoud@yahoo.fr}


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