Collineation groups with perspectivities.
Let \({\mathcal T}\) be a finite translation plane. The collineation group of \({\mathcal T}\) is a semi-direct product of the translation group and the translation complement. The translation complement is a semi-linear transformation group. A subgroup of the translation complement consisting of linear collineations is called a linear collineation group of \({\mathcal T}\). In the paper under review Ho provides some remarkable results about linear collineation groups of finite translation planes. His main results read as follows. Theorem 1. Let \(G\) be a simple linear collineation group containing a perspectivity of a finite translation plane. (i) If \(G\) is isomorphic to an alternating group \(A_n\), then \(n=5\), and the translation plane is of even order. (ii) \(G\) is not isomorphic to any one of the 26 sporadic finite simple groups. Theorem 2. Let \(G\) be a simple linear collineation group containing an elation of a finite translation plane. (i) The translation plane is of even order and \(G \cong L_2(u)\) or \(G \cong Sz(u)\), where \(u\) is a power of 2. (ii) If all involutions of \(G\) are Baer involutions, then any perspectivity in \(G\) has an order dividing \(u + 1\).
scientific article; zbMATH DE number 2066832