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 EMS Series of Lectures in Mathematics 2012; 129 pp; softcover Volume: 17 ISBN-10: 3-03719-110-4 ISBN-13: 978-3-03719-110-1 List Price: US$36 Member Price: US$28.80 Order Code: EMSSERLEC/17 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. Table of Contents Generalized quadrangles The Moufang condition Elation quadrangles Some features of special $$p$$-groups Parameters of elation quadrangles and structure of elation groups Standard elations and flock quadrangles Foundations of EGQs Elation quadrangles with nonisomorphic elation groups Application: Existence of translation nets Elations of dual translation quadrangles Local Moufang conditions Bibliography Symbols Index