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%  Groups in Action, Thurnau, Oktober 1996
%  ABSTRACT. Symmetry Groups of Finite Generalized Quadrangles.
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{\Large \bf         Symmetry Groups of Finite Generalized Quadrangles. }

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Dirk Hachenberger \\
Institut f\"ur Mathematik, Universit\"at Augsburg \\
D-86159 Augsburg \\
{\footnotesize Hachenberger@math.uni-augsburg.de}
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{\bf Abstract.}
Finite Geometries are often studied under the assumption of certain automorphism groups,
whence examples can be constructed by coordinatizing the underlying object in terms of the
group acting on it. This is also the case for class of geometries,
I am considering in my talk, namely Finite Generalized Quadrangles (GQ).

GQs were introduced as particular Generalized Polygones by J. Tits (1959), and have attained much
interest since then. The standard reference is the monography
of S. Payne and J. Thas (1984). Most of the presently known families of GQs
can be represented as a group coset geometry with respect to a certain group of
automorphisms. Necessarily, the subgroup lattice of the coordinatizing group has to satisfy
a particular combinatorial property (it has to admit a so-called Kantor family or 4-gonal family).

We report on recent results on groups admitting a Kantor family under the assumption that the
family contains some
particular symmetries of the coordinatized quadrangle.
One of our main results is that the parameters of a Skew Translation Generalized Quadrangle
(STGQ) necessarily are powers of the same prime.
This is of interest since all the nine new families of GQs found after
1980 belong to the class of STGQ. Moreover, it settles the prime power conjecture of W. Kantor
for the class of STGQ.

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