# A Course on Elation Quadrangles

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*Koen Thas*

A publication of the European Mathematical Society

The notion of elation generalized quadrangle is a natural
generalization to the theory of generalized quadrangles of the important notion
of translation planes in the theory of projective planes. Almost any known
class of finite generalized quadrangles can be constructed from a suitable
class of elation quadrangles.

In this book the author considers several aspects of the theory of
elation generalized quadrangles. Special attention is given to local
Moufang conditions on the foundational level, exploring, for instance,
Knarr's question from the 1990s concerning the very notion of elation
quadrangles. All the known results on Kantor's prime power conjecture
for finite elation quadrangles are gathered, some of them published
here for the first time. The structural theory of elation quadrangles
and their groups is heavily emphasized. Other related topics, such as
\(p\)-modular cohomology, Heisenberg groups, and existence
problems for certain translation nets, are briefly touched.

This book starts from scratch and is essentially
self-contained. Many alternative proofs are given for known
theorems. This course contains dozens of exercises at various levels,
from very easy to rather difficult, and will stimulate undergraduate
and graduate students to enter the fascinating and rich world of
elation quadrangles. More accomplished mathematicians will find the
final chapters especially challenging.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Undergraduate and graduate students and research mathematicians interested in elation quadrangles.