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\begin{document}

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\titleofpresentation{Abelian normal subgroups of unipotent subgroups  \\
of Chevalley groups }

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

%  [Body of abstract goes here.]

Throughout this paper $G(K)$ denotes a Chevalley  group over an
arbitrary field $K$, associated with the root system $G$. In this
work the maximal abelian normal subgroups of the unipotent
subgroup $UG(K)$ are described. The terminology from
\cite{suleymanova2,suleymanova1,suleymanova4} will be used. The
description was partially announced in \cite{suleymanova5}. For
the classical types $G=A_n,\ B_n,\ C_n,\ D_n,\ ^2A_m,\ ^2D_{n+1}$
the results were obtained with V.M. Levchuk.

For example, the following theorem is proofed:

%\medskip

{\bf Theorem 1.} \emph { Suppose that $M$ is a maximal abelian
normal subgroup of $UG(K)$.}

{\it a) If $G={}^2B_2$, then $M=Z_1 <\alpha>$ with an arbitrary
$\alpha \not\in Z_1$;}

{\it b) If $G=G_2$, $3K=0$, or $G=\ ^2G_2$, then $M=Z_2=U_2$;}

{\it c) If $G=G_2$, $2K=0$ and $|K|>2$, then either
$M=\{x_{a+b}(t)x_{2a+b}(td) \ | \ t\in K\}Z_2$ with an arbitrary
$d \in K$ and $Z_2=U_4$, or $M=U_3;$}

{\it d) If $G=G_2$, $|K|=2$, then $M= \langle
x_a(1)x_{2a+b}(1)\rangle Z_2$, $Z_2=\ \langle
x_{a+b}(1)x_{2a+b}(1) \rangle U_4$, or $M=U_3$, or $M=
X_{a+b}U_4$;}

{\it e) If $G=G_2, \ 6K=K$, or $G= \ ^3D_4$, then $M=Z_3=U_3$;}

{\it f) If $G={}^2F_4$, then $M=\{R_{3,-2}(t)R_{42}(td) \ | \ t\in
K \}Z_4$ with an arbitrary  $d \in K$ or $M=<R_{43}(a)>
R_{42}(K)Z_4$, $a \in K^*$.}

%\medskip

The full description of the set $A_N(U)$ of large abelian normal
subgroups of $U=UG(K)$ over the finite field $K=GF(q)$
\cite[Problem (1.6)]{suleymanova3}) may be obtained from these
results. For example, the next list of subgroups from $A_N(U)$ and
their order $a(U)$ may be obtained from Theorem 1:

%\medskip

{\bf  Table 2. }

%\bigskip
\noindent
\begin{tabular}{|c|l|l|l|}

\hline

$G$ & $A_N(U)$ & $a(U)$ & $K$ \\

\hline

${}^2B_2(q)$ & $Z_1 <\alpha>$ \ \ ($\alpha \not\in Z_1$)& $2q$ &  \\

\hline

${}^2G_2(q)$ & $U_2$ & $q^2$ & \\

\hline

$^3D_4(q^3)$ & $U_3$ & $q^5$ & \\

\hline

$G_2(q)$ & $U_3$ & $q^3$ & $6K=K$\\

& $U_2$ & $q^4$ & $3K=0$\\

& $U_3,\ \{x_{a+b}(t)x_{2a+b}(td) \ | \ t\in K\}U_4$ \ \ ($|K|>2$) & $q^3$ & $2K=0$ \\

& $\langle x_a(1)x_{2a+b}(1),\  x_{a+b}(1)x_{2a+b}(1) \rangle U_4$ & $q^4$ & $|K|=2$ \\

\hline

$^2F_4(q)$ &  $<R_{43}(a)> R_{42}(K)U_5$ & $2q^5$ &\\

\hline

\end{tabular}\\
\\

This research is supported by Russian fund of Fundamental
researches, grant No. 06-01-00824a.


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\bibitem{suleymanova1} N. Bourbaki, {\it Groupes et algebres de Lie (IV--VI)}, Hermann, Paris, 1968.

\bibitem{suleymanova2} R. Carter, {\it Simple groups of Lie type}, New York, Wiley and Sons, 1972.

\bibitem{suleymanova3} A. S. Kondratyev, {\it Subgroups of finite Chevalley groups, }Uspekhi
Mat. Nauk, {\bf 41}:1(247) (1986), 57-96.

\bibitem{suleymanova4} V.M. Levchuk, {\it Automorphisms of unipotent subgroups of
Chevalley groups}, Algebra and Logic, {\bf 29}:2 (1990), 141-161
(Part. I); {\bf 29}:3, 315-338 (Part. II).

\bibitem{suleymanova5} V.M. Levchuk, {\it Generalized unipotent subgroups of classical
groups}, Fund. and Appl. Math., {\bf 2}:2 (1996), 625-627.

\end{ourbiblio}

\presenteraddress{Siberian Federal University}  % The name of a university is fine.

\email{suleymanova@list.ru}

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