By Koen Thas

The suggestion of elation generalized quadrangle is a typical generalization to the speculation of generalized quadrangles of the $64000 thought of translation planes within the thought of projective planes. nearly any recognized classification of finite generalized quadrangles should be created from an appropriate category of elation quadrangles.

In this ebook the writer considers numerous facets of the idea of elation generalized quadrangles. specific cognizance is given to neighborhood Moufang stipulations at the foundational point, exploring for example a question of Knarr from the Nineties about the very proposal of elation quadrangles. all of the identified effects on Kantor’s leading strength conjecture for finite elation quadrangles are accumulated, a few of them released right here for the 1st time. The structural idea of elation quadrangles and their teams is seriously emphasised. different similar themes, 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 recognized theorems. Containing dozens of routines at quite a few degrees, from really easy to particularly tricky, this path will stimulate undergraduate and graduate scholars to go into the interesting and wealthy global of elation quadrangles. The extra entire mathematician will in particular locate the ultimate chapters hard.

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0; 0; 0; 1/ of K. Such a plane i has equation ai X0 Cbi X1 Cci X2 CX3 D 0. ci cj /X2 D 0. Let q be odd. bi bj / is a nonsquare. bi bj / is a nonsquare, whenever i ¤ j . Let q be even. ci C cj / 2 2 C1 . ci C cj / 2 2 C1 , whenever i ¤ j. 2. 0; 0; 0/. q/. v1 ; v2 / 2 Fq2 , then put u v D u1 v1 C u2 v2 . 0; 0/). 2 2/-matrix over Fq , with the convention that A0 be the zero matrix. t / C , t 2 Fq [ f1g. Fq [ f1g. t/ is a commutative subgroup of G having order q 3 , t 2 Fq [ f1g. t/. t / j t 2 Fq [ f1gg: With the foregoing notations we have the following two important theorems.

5 (J. A. Thas [52]). q 2 ; q/ if and only if the planes x t X0 C z t X1 C y t X2 C X3 D 0, t 2 Fq , define a flock F of the quadratic cone with equation X0 X1 D X22 . G; J/ of the type described above gives us a flock of the quadratic cone. F / and is called a flock GQ. 2 Linear flocks. A flock F is linear if all the flock planes contain a common line. The interest in linear flocks is reflected in the next characterization of classical flock GQs. 6 (J. A. Thas [52]). 3; q 2 /. We have introduced flock GQs as a particular class of EGQs.

So either G is a p-group, or X is a Sylow p-subgroup of G. 8 led D. Hachenberger to prove a well-known conjecture of S. E. 9 (Hachenberger [21]). The parameters of any thick finite STGQ are powers of one and the same prime. Proof. 1; t/. 8 (a). 48 5 Parameters of elation quadrangles and structure of elation groups In the next section, we will give another proof of this result. In [21] D. 8 cannot occur. In [74], we “completed” his classification by proving that this conjecture is indeed true. While I was writing up the present manuscript, I was not able to reconstruct the combinatorial lemma (on subquadrangles) stated in [74] (erroneously) without proof.

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