A Course on Elation Quadrangles by Koen Thas

By Koen Thas

The thought of elation generalized quadrangle is a usual generalization to the idea of generalized quadrangles of the $64000 idea of translation planes within the idea of projective planes. nearly any identified type of finite generalized quadrangles will be created from an appropriate type of elation quadrangles.

In this e-book the writer considers numerous elements of the speculation of elation generalized quadrangles. exact realization is given to neighborhood Moufang stipulations at the foundational point, exploring for example a query of Knarr from the Nineties in regards to the very concept of elation quadrangles. the entire identified effects on Kantor’s best strength conjecture for finite elation quadrangles are accumulated, a few of them released right here for the 1st time. The structural thought of elation quadrangles and their teams is seriously emphasised. different comparable subject matters, resembling p-modular cohomology, Heisenberg teams and life difficulties for convinced translation nets, are in brief touched.

The textual content begins from scratch and is basically self-contained. many different proofs are given for recognized theorems. Containing dozens of routines at quite a few degrees, from really easy to relatively tricky, this direction 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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Fong and G. M. Seitz determined all finite split BN-pairs of rank 2 (the B2 -case being, by far, the most complicated type to handle). We only state the result for BN-pairs of type B2 . 4 (Fong and Seitz [16], [17]). Let G be a finite group with an effective, saturated split BN-pair of rank 2 of type B2 . q/. Equivalently, a thick finite generalized quadrangle is isomorphic, up to duality, to one of the classical examples if and only if it verifies the Moufang Condition. Much more recently the conditions of the previous theorem were relaxed still to obtain the same conclusion.

E AA AA }} } AA } AA }} } } /A / 0. 0 G AA }> } AA } AA }} A  }}} E0 We also mention the following result on the second cohomology group. 5. 0 with a given G-module structure on A. G; A/. G; A/. G; A/. Exercise. Let G be a finite group and A be a G-module. G; A/, n 2 N, has a finite order which divides jGj. G; A/ D 0 for all n 1. So any extension of G by A is split. P / is elementary abelian or P itself is. Note that P =ŒP; P  is elementary abelian in that case. P /j D p, P is called extra-special.

It is the “odd one out” chapter of this book. n C 2/-matrices with entries in Fq , of the following form (and with the usual matrix multiplication): 0 1 1 ˛ c @0 In ˇ T A ; 0 0 1 where ˛; ˇ 2 Fqn , c 2 Fq and with In being the n n-unit matrix. y1 ; y2 ; : : : ; yn / elements of Fqn , denotes x1 y1 C x2 y2 C C xn yn D xy T . 1. ˛ C ˛ 0 ; c C c 0 C ˛ˇ 0 ; ˇ C ˇ 0 /: T Throughout these notes, we keep using the latter representation for the general Heisenberg group. Below, 0N denotes the null vector of Fqn .

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