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A course on elation quadrangles. (English) Zbl 1253.51005

EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-110-1/pbk). xii, 116 p. (2012).
In this book, the author studies finite elation generalized quadrangles (EGQs). The book contains virtually all known results about these structures up to now. Moreover, some related things are briefly touched, such as group cohomology, Lie algebras and extraspecial \(p\)-groups (although one can argue that it is too brief, and that some motivation is lacking).
An EGQ is a generalized quadrangle, together with a distinguished point \(x\), the elation point, such that some group of automorphisms fixing all lines through \(x\) acts simply transitively on the set of points not collinear with \(x\). The theory of EGQs is important in that almost every known finite generalized quadrangle is, up to duality, an EGQ. The exceptions are all constructed from an EGQ. Hence one can say that the study of EGQs is equivalent with the study of all known finite generalized quadrangles. A systematic theory of finite EGQs was started by S. E. Payne and J. A. Thas in their book [Finite generalized quadrangles. EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (2009; Zbl 1247.05047)]. The most important results of the last decade are due to K. Thas, the author of the book under review.
The highlights of these results, in my view, are an affirmative answer to a question of Kantor, and the solution of Knarr’s problem, both contained in detail in the book. Kantor’s question asks whether the dual of a non-classical “good” translation generalized quadrangle of order \((q,q^2)\) always has an elation point. The author proves that it always has a unique elation point and a unique elation group. This is discussed in Chapter 10 of the current book. Knarr’s problem is related to the Moufang condition, which requires that for every triplet of elements \(a,b,c\), where \(a\) is incident with \(b\) and \(b\) is incident with \(c\neq a\), the group of automorphisms (called root-elations) fixing all elements incident with one of \(a,b,c\) acts transitively on the elements different from \(a\) incident with an arbitrary element incident with \(a\). The local Moufang condition at a point \(x\) requires that this holds for every triplet containing \(x\) not as middle element. Knarr’s question was whether this local Moufang condition at \(x\) implies that \(x\) is an elation point with elation groups the group generated by all corresponding root-elations. The author proves in Chapter 11 that this is indeed true, and provides a simplified proof in the book. Moreover, he shows that the elation group is a \(p\)-group in this case, which is a rather strong result.
These results show that the author is the world expert on finite elations quadrangles, and he now sets the beacons for further research. In this respect, it would have been good to include some hints about where the theory should go from here. The full prime-power conjecture for EGQs seems to be a very hard problem, but there are certainly other interesting problems, conjectures and applications in this nice theory. Anyway, the present book is a very nice presentation of the subject and it should motivate more people to work in this beautiful area.

MSC:

51E12 Generalized quadrangles and generalized polygons in finite geometry
51B25 Lie geometries in nonlinear incidence geometry
05B25 Combinatorial aspects of finite geometries
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures

Citations:

Zbl 1247.05047
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