By Koen Thas
The thought of elation generalized quadrangle is a ordinary generalization to the speculation of generalized quadrangles of the $64000 idea of translation planes within the concept of projective planes. virtually any identified classification of finite generalized quadrangles might be produced from an appropriate classification of elation quadrangles.
In this publication the writer considers a number of facets of the idea of elation generalized quadrangles. exact consciousness is given to neighborhood Moufang stipulations at the foundational point, exploring for example a query of Knarr from the Nineteen Nineties about the very thought of elation quadrangles. all of the recognized effects on Kantor’s top strength conjecture for finite elation quadrangles are accumulated, a few of them released right here for the 1st time. The structural concept of elation quadrangles and their teams is seriously emphasised. different comparable issues, equivalent to p-modular cohomology, Heisenberg teams and life difficulties for convinced translation nets, are in short touched.
The textual content starts off from scratch and is largely self-contained. many different proofs are given for identified theorems. Containing dozens of workouts at quite a few degrees, from really easy to quite tricky, this path will stimulate undergraduate and graduate scholars to go into the attention-grabbing and wealthy global of elation quadrangles. The extra comprehensive mathematician will particularly locate the ultimate chapters not easy.
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Additional resources for A Course on Elation Quadrangles
An extra-special p-group P is the central product of r nonabelian subgroups of order p 3 . Moreover, we have 1 (1) If p is odd, P is isomorphic to N k M r k , while if p D 2, P is isomorphic to D k Qr k for some k. In either case, jP j D p 2rC1 . (2) If p is odd and k 1, N k M r k is isomorphic to NM r NM r 1 are not isomorphic and M r is of exponent p. 1 , the groups M r and (3) If p D 2, then D k Qr k is isomorphic to DQr 1 if k is odd and to Qr if k is even, and the groups Qr and DQr 1 are not isomorphic.
A. Thas ). 3; q 2 /. We have introduced flock GQs as a particular class of EGQs. As the following result shows, in general the elation point is unique. 7 (Payne and J. A. Thas ). F / has one and only one elation point. F /; if F is not linear, this refers to the unique elation point. If F is linear, all points are special. 2 Fundamental theorem of q-clan geometry In this section, we describe the so-called “Fundamental theorem of q-clan geometry”. Ci //, i D 1; 2. 1 The fundamental theorem.
We leave the explicit calculations to the reader. 2 Elation generalized quadrangles We have observed that all finite classical GQs and their point-line duals have, for each point, an automorphism group that fixes it linewise and has a sharply transitive action on the points which are noncollinear with that point. 1 Elations and quadrangles. P ; B; I/ be a GQ. If there is an automorphism group H of Ã which fixes some point x 2 P linewise and acts sharply transitively on P n x ? , we call x an elation point, and H “the” associated elation group.