\titleofpresentation{Locally solvable$^\circ$ groups of finite Morley rank}

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

%  [Body of abstract goes here.]
This is joint work with Eric Jaligot.
The main conjecture concerning groups of finite Morley rank is the
Cherlin--Zilber one, stating that any simple infinite group of finite
Morley rank is isomorphic to an algebraic group over an algebraically
closed field. A proof of (some reasonable part of) this conjecture is
expected to run parallel to the classification of finite simple
groups, and vice-versa. 

As there is no Feit--Thompson theorem for groups of finite Morley rank,
one has to assume there are involutions. Now as in the finite case, it
is natural to consider at some point ``locally solvable groups''. This
amounts to transfering results obtained on so-called ``minimal
connected simple groups'' to a larger class. \emph{Locally} here does
not mean finitely generated, but comes from the following. 

\begin{definition}
Let $G$ be a group. A subgroup $H \leq G$ is said to be \emph{local} if there is some non-trivial abelian subgroup $A \leq G$ such that $H \leq N_G (A)$.
\end{definition}

\begin{definition}
Let $G$ be a group of finite Morley rank. $G$ is said to be \emph{locally solvable} if for any non-trivial abelian subgroup $A \leq G$, $N_G (A)$ is solvable.
\end{definition}

As people usually work with connected components (denoted by a $^\circ$), our definition is the following.

\begin{definition}
Let $G$ be a group of finite Morley rank. $G$ is said to be \emph{locally solvable$^\circ$} if for any non-trivial abelian subgroup $A \leq G$, $N_G^\circ (A)$ is solvable.
\end{definition}

The only non-solvable locally solvable$^\circ$ algebraic group is $\rm PSL_2 (K)$, which does have involutions.

Our main tools rely on a notion of unipotence in characteristic $0$ developed by Jeffrey Burdges in his PhD \cite{DeloroBSimple}.
Namely one has to generalize the notion of a characteristic in the following way.

\begin{definition}
A \emph{unipotence parameter} is a pair $\tilde{q} = (p, d)$, where $p$ is either prime or $\infty$, $d \in \mathbb{N} \cup \{\infty\}$, such that $p \neq \infty \Leftrightarrow d = \infty$.
\end{definition}

So if $p$ is a real prime number, this is just indicating the
characteristic, but if $p = \infty$ the notion of unipotence becomes
graded ($d$ is called the unipotence degree). In the latter case
Burdges defined $\tilde{q}$-unipotent subgroups, and $\tilde{q}$-Sylow
subgroups. As far as results and not definitions are concerned, both
these notions provide good analogs of the prime characteristic ones. 

There is an important uniqueness lemma that was first proved in the
minimal connected simple framework \cite{DeloroGroupes1}. 

\begin{lemma}[Uniqueness Lemma]
Let $G$ be a locally solvable$^\circ$ group if finite Morley rank. Let $\tilde{q} = (p, d)$ be some unipotence parameter with $d > 0$, and let $U$ be some $\tilde{q}$-Sylow subgroup of $G$.

Assume $U$ contains some $\tilde{q}$-unipotent subgroup $U_1$, and that $U_1$ contains some subset $X$ such that the $p$-unipotence degree of $C^\circ (X)$ is not bigger than $d$.

Then $U$ is the only $\tilde{q}$-Sylow subgroup of $G$ containing $U_1$.
\end{lemma}


This lemma is strong enough to provide a good intersection control, allowing us to prove the following.

\begin{theorem*}[Mixed type theorem]
Let $G$ be a locally solvable$^\circ$ group of finite Morley rank. Assume $G$ has mixed type. Then $G^\circ$ is solvable 
\end{theorem*}

\begin{theorem*}[Even type theorem]
Let $G$ be a locally solvable$^\circ$ group of finite Morley rank. Assume $G$ has even type. Then either $G^\circ$ is solvable or $G \simeq \rm PSL_2 (K)$ for some algebraically closed field $K$ of characteristic $2$.
\end{theorem*}

Our results in odd type are not as positive yet.

\begin{theorem*}[Odd type algebraic theorem]
Let $G$ be a locally solvable$^\circ$ group of finite Morley
rank. Assume $G$ has odd type and Pruefer-rank $1$. Assume also that
there is a toral involution $i$ such that $C^\circ (i)$ is not a Borel
subgroup. Then either $G^\circ$ is solvable or $G \simeq \rm PSL_2
(K)$ for some algebraically closed field $K$ of characteristic not
$2$. 
\end{theorem*}

\begin{theorem*}[Odd type non-algebraic theorem]
Let $G$ be a non-solvable locally solvable$^\circ$ group of finite
Morley rank. Assume $G$ has odd type and Pruefer-rank $\geq 2$. Then
$G$ has Pruefer-rank equal to $2$, its Sylow subgroup is connected,
and involutions are conjugate. Furthermore for any involution $i$ of
$G$, $C^\circ (i)$ is a Borel subgroup. 
\end{theorem*}


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

% \bibitem[BBC06]{BBCInvolutions}
% Alexandre Borovik, Jeffrey Burdges, and Gregory Cherlin.
% \newblock Involutions in groups of finite {M}orley rank.
% \newblock 2006.
% \newblock Soumis.
% 
% \bibitem[BC06]{BCOn}
% Jeffrey Burdges and Gregory Cherlin.
% \newblock On semisimple torsion in groups of finite {M}orley rank.
% \newblock 2006.
% \newblock Preprint.
% 
% \bibitem[BC07]{BCPermutation}
% Alexandre Borovik and Gregory Cherlin.
% \newblock Permutation {G}roups of finite {M}orley rank.
% \newblock {\em {C}ambridge {U}niversity {P}ress - {N}ewton {I}nstitute}, 2007.
% \newblock A para�re.
% 
% \bibitem[BCJ07]{BCJMinimal}
% Jeffrey Burdges, Gregory Cherlin, and Eric Jaligot.
% \newblock Minimal connected simple groups of finite {M}orley rank with strongly
%   embedded subgroups.
% \newblock {\em J. Algebra}, 2007.
% \newblock A para�re.
% 
% \bibitem[BN94]{BNGroups}
% Alexandre Borovik and Ali Nesin.
% \newblock {\em Groups of finite {M}orley rank}, volume~26 of {\em Oxford Logic
%   Guides}.
% \newblock The Clarendon Press Oxford University Press, New York, 1994.
% \newblock Oxford Science Publications.
% 
% \bibitem[Bor95]{BSimplelocally}
% Alexandre Borovik.
% \newblock Simple locally finite groups of finite {M}orley rank and odd type.
% \newblock In {\em Finite and locally finite groups (Istanbul, 1994)}, volume
%   471 of {\em NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.}, pages 247--284.
%   Kluwer Acad. Publ., Dordrecht, 1995.
% 
% \bibitem[BP90]{BPTores}
% Aleksandr~Vasilievich Borovik and Bruno~Petrovich Poizat.
% \newblock Tores et {$p$}-groupes.
% \newblock {\em J. Symbolic Logic}, 55(2):478--491, 1990.
% 
\bibitem[Bur04]{DeloroBSimple}
Jeffrey Burdges.
\newblock {\em Simple Groups of Finite {M}orley Rank of Odd and Degenerate
  Type}.
\newblock PhD thesis, Rutgers University, New Brunswick, New Jersey, 2004.
 
 \bibitem[Bur07]{DeloroBBender}
 Jeffrey Burdges.
 \newblock The {B}ender method in groups of finite {M}orley rank.
 \newblock {\em J. Algebra}, 2007.
 \newblock To appear.
% 
% \bibitem[Che05]{CGood}
% Gregory Cherlin.
% \newblock Good tori in groups of finite {M}orley rank.
% \newblock {\em J. Group Theory}, 8(5):613--621, 2005.
% 
% \bibitem[CJ04]{CJTame}
% Gregory Cherlin and Eric Jaligot.
% \newblock Tame minimal simple groups of finite {M}orley rank.
% \newblock {\em J. Algebra}, 276(1):13--79, 2004.
% 
 \bibitem[Del07a]{DeloroGroupes1}
 Adrien Deloro.
 \newblock Groupes simples connexes minimaux alg\'{e}briques de type impair.
 \newblock {\em Journal of Algebra}, 2007.
 \newblock To appear.

 \bibitem[Del07b]{DeloroGroupes2}
 Adrien Deloro.
 \newblock Groupes simples connexes minimaux non-alg\'{e}briques de type impair.
 \newblock Submitted, 2007.

% 
% \bibitem[FJ07]{FJConjugacy}
% Olivier Fr�on and Eric Jaligot.
% \newblock Conjugacy in groups of finite {M}orley rank.
% \newblock {\em {C}ambridge {U}niversity {P}ress - {N}ewton {I}nstitute}, 2007.
% \newblock A para�re.
% 
% \bibitem[Jal06]{JGenerix}
% Eric Jaligot.
% \newblock Generix never gives up.
% \newblock {\em J. Symbolic Logic}, 71(2):599--610, 2006.

\end{ourbiblio}

\presenteraddress{Paris 7 University}  % The name of a university is fine.

\email{adeloro@logique.jussieu.fr}

%\web{}  % Those without homepages should delete this.

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