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Point-primitive, line-transitive generalised quadrangles of holomorph type. (English) Zbl 1428.20004

Summary: Let \(G\) be a group of collineations of a finite thick generalised quadrangle \(\Gamma\). Suppose that \(G\) acts primitively on the point set \({\mathcal{P}}\) of \(\Gamma\), and transitively on the lines of \(\Gamma\). We show that the primitive action of \(G\) on \({\mathcal{P}}\) cannot be of holomorph simple or holomorph compound type. In [the first author et al., J. Comb. Des. 24, No. 4, 151–164 (2016; Zbl 1338.05031)], we have previously classified the examples \(\Gamma\) for which the action of \(G\) on \({\mathcal{P}}\) is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup \(M\) and \(M\) is non-Abelian.

MSC:

20B15 Primitive groups
05B25 Combinatorial aspects of finite geometries
51E12 Generalized quadrangles and generalized polygons in finite geometry
51E14 Finite partial geometries (general), nets, partial spreads

Citations:

Zbl 1338.05031
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References:

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