SEMINAR: Groups and Combinatorics Seminar: Generalised quadrangles and simple groups


Groups and Combinatorics Seminar: Generalised quadrangles and simple groups 
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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 pointregular 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 pointregular group of automorphisms necessarily arises from a socalled pseudohyperoval in a certain projective space. In particular, there are many examples of such GQs. A natural question in the nonabelian case is the following. Let T be a nonabelian 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 pointprimitive automorphism group of O’NanScott type HS (r=1) or HC (r>1). The latter situation has previously been shown to be impossible under certain additional assumptions: first lineprimitivity (Bamberg, Giuidici, Morris, Royle & Spiga, 2012), and then linetransitivity (Bamberg, Popiel & Praeger, to appear). As part of a longterm project to classify the pointprimitive 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 nearresult and, time permitting, summarise our progress on the overarching classification project. This is joint work with John Bamberg and Cheryl Praeger.
Contact 
Tomasz Popiel
<[email protected]>

Start 
Fri, 21 Oct 2016 16:00

End 
Fri, 21 Oct 2016 17:00

Submitted by 
Tomasz Popiel <[email protected]>

Last Updated 
Mon, 24 Oct 2016 10:30

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