SEMINAR: Groups and Combinatorics Seminar: Generalised quadrangles and simple groups
|Groups and Combinatorics Seminar: Generalised quadrangles and simple groups
Time and place: 16:00 Friday 21 October in Weatherburn LT
Speaker: Tomasz Popiel (UWA)
Title: Generalised quadrangles and simple groups
Abstract: Generalised quadrangles (GQs) with point-regular automorphism groups have been studied since at least 1992, when Dina Ghinelli conjectured (but fell slightly short of proving) that if a GQ of order (s,t) admits a group of automorphisms acting regularly on points, then s=t. The abelian case is well understood, in the sense that a GQ with an abelian point-regular group of automorphisms necessarily arises from a so-called pseudo-hyperoval in a certain projective space. In particular, there are many examples of such GQs. A natural question in the non-abelian case is the following. Let T be a non-abelian finite simple group and r a positive integer, and consider the group M=T^r acting on itself by both right and left multiplication. Can there be a generalised quadrangle Q with point set M that admits such a group? This scenario arises, in particular, if we assume that Q admits a point-primitive automorphism group of O’Nan-Scott type HS (r=1) or HC (r>1). The latter situation has previously been shown to be impossible under certain additional assumptions: first line-primitivity (Bamberg, Giuidici, Morris, Royle & Spiga, 2012), and then line-transitivity (Bamberg, Popiel & Praeger, to appear). As part of a long-term project to classify the point-primitive GQs, we have almost managed to show that, in fact, no symmetry assumption on lines is necessary. As of this writing, it is known that r must equal 1, and that T must be one of 17 groups of Lie type (each of Lie rank at most 7). I will sketch the proof of this near-result and, time permitting, summarise our progress on the overarching classification project. This is joint work with John Bamberg and Cheryl Praeger.
Fri, 21 Oct 2016 16:00
Fri, 21 Oct 2016 17:00
Tomasz Popiel <[email protected]>
Mon, 24 Oct 2016 10:30
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