\titleofpresentation{Linear groups acting transitively and
  flag-transitively on projective spaces} 

\presenter{Sergey Zyubin}

Let $K$ be a (commutative or skew) field and $PG(n,K)$ be a projective space over $K$.
The projective linear group $PGL_{n+1}(K)$ naturally acts on $PG(n,K)$.
Under such an action, the classical group $PSO_{n+1}(\mathbb{R})$ over the real field $\mathbb{R}$
acts transitively on the projective space $PG(n,\mathbb{R})$.
Moreover it acts flag-transitively, i.e. it moves a maximal flag of the space to any other one.
Another example of transitive subgroup of the projective linear group
is the subgroup $PSL_2(\mathbb{Z})$ of the group $PGL_2(\mathbb{Q})$.
The following two problems are part of the problem 11.70 from the Kourovka Notebook.
This problem was posed by P. Neumann and Ch. Praeger for infinite fields.
(1) Find all subgroups of $PGL_2(K)$, which act transitively on the projective line $PG(1,K)$;
(2) under what conditions does the subgroup $PGL_{n+1}(R)$ over a subring $R$ of the field $K$ 
    act flag-transitively on $PG(n,K)$?

The next theorem gives answer for the second part of the problem for commutative fields. 
 
\begin{theorem}
Let $R$ be a subring of a (commutative) field $K$.
Then the subgroup $PGL_{n+1}(R)$ acts flag-transitively on the projective space $PG(n,K)$
if and only if $R$ is a Bezout ring and its quotient field coincides with $K$.
\end{theorem}
 
The second result gives answer for the first part of the problem for locally finite fields.

\begin{theorem}
Let $K$ be a locally finite field and $G$ be a subgroup of $PGL_2(K)$.
If $G$ acts transitively on the projective line $P(1,K)$ then the only following cases are possible:
\begin{enumerate}
\item  
$G=PGL_2(K)$;
\item
$G=PSL_2(K)$;
\item
 $G$ is conjugated over quadratic extension $^2K$ of $K$ to the monomial subgroup over $^2K$;
\item
$G$ has index 2 in the subgroup from the previous case. 
\end{enumerate}
\end{theorem}

{This research is supported by the Russian Foundation for Basic Research (Grant 06--01--00824).}

\presenteraddress{Tomsk Polytechnic University,
av. Lenina, 30,
634050, Tomsk,
Russia}

\email{sergey.zyubin@gmail.com}

\web{www.genealogy.ams.org/html/id.phtml?id=65105}



