On generalized quadrangles with some concurrent axes of symmetry
The author basically considers the following question: Given a generalized quadrangle \({\mathcal S}\), a point \(x\) in it and \(k\) axes of symmetry passing through \(x\) (an axis of symmetry \(L\) is a line such that the group of automorphisms fixing all lines concurrent with \(L\) has order \(s\), with \((s, t)\) the order of \({\mathcal S}\), i.e., every line of \(S\) carries exactly \(1+s\) points, and every point is incident with precisely \(1+t\) lines), under which additional conditions can we conclude that (a) all lines through \(x\) are axes of symmetry, (b) the related symmetry groups generate the symmetry groups corresponding to all lines through \(x\). The author offers a wealth of answers, going from using additional combinatorial properties, to using only a numerical bound on \(k\). It is also explained how his investigation relates with flock generalized quadrangles, and new characterizations emerge. In fact, there are so many results mentioned and proved in this paper, which relate to so many different aspects, that it is not easy to have an overview of them after a first reading. Personally, I most like the last main result stating that, with the above notation, if \(t\geq s\neq 1\) and \(k> t- s+2\), then every line through \(x\) is an axis of symmetry, and every symmetry group is generated by \(t- s+ 2\) arbitrary symmetry groups (provided \(s\neq t\); if \(s= t\), then one needs 3 symmetry groups).
scientific article; zbMATH DE number 2031081