Simon Thomas

Mathematics Department, Rutgers University, 110 Frelinghuysen Road
Piscataway, New Jersey 08854-8019
e-mail: sthomas@math.rutgers.edu

Title: The automorphism tower problem


If $G$ is a centreless group, then there is a natural embedding  
of $G$ into its automorphism group $Aut(G)$, obtained by sending 
each $g\in G$ to the corresponding inner automorphism 
$i_{g} \in Aut(G)$. It turns out $Aut(G)$ is also centreless, and
this enables us to define the automorphism tower of $G$ to be the
ascending chain of groups
\[
G = G_{0} \leqslant G_{1} \leqslant G_{2} \leqslant  \cdots
G_{\alpha} \leqslant G_{\alpha +1} \leqslant \cdots
\]
such that for each ordinal $\alpha$
\begin{enumerate}
\item[(a)] $G_{\alpha +1} = Aut(G_{\alpha})$; and
\item[(b)] if $\alpha$ is a limit ordinal, then
$G_{\alpha} = {\bigcup}_{\beta < \alpha} G_{\beta}$.
\end{enumerate}
The automorphism tower is said to terminate if there exists an ordinal
$\alpha$ such that $G_{\alpha +1} = G_{\alpha}$. In 1939, Wielandt
proved that if $G$ is finite, then its automorphism tower terminates 
after finitely many steps. In 1984, I showed that the automorphism tower 
of an arbitrary centreless group eventually terminates; and that for each 
ordinal $\alpha$, there exists a group whose automorphism tower terminates 
after exactly $\alpha$ steps.

In this talk, I will survey some more recent developments in this area
and discuss some of the many open problems.