Generalised quadrangles VI: translation generalised quadrangles

I am finally onto my last post on generalised quadrangles. The topic of this one is translation generalised quadrangles. These are a special case of elation generalised quadrangles outlined in my last post. The main source for this post has been Michel Lavrauw’s Phd thesis which is availiable from Ghent’s PhD theses in finite geometry page.

Recall that an elation generalised quadrangle (EGQ) is a generalised quadrangle with a base point {x} and a group of automorphisms that fixes each line incident with {x} and acts regularly on the set of points not collinear with {x}. A translation generalised quadrangle (or TGQ) is an EGQ with an abelian elation group. In this case the elation group is referred to as a translation group. We saw in the EGQ post, that {\mathsf{W}(3,q)}, for {q} even, has an abelian elation group and so is an example of a TGQ. It has been proved by Stan Payne and Jef Thas that for a TGQ, the elation group must be elementary abelian and in particular {s} and {t} are powers of the same prime.

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