Finite Minkowski planes and embedded inversive planes (Q5945003)

In [J. Geom. 41, 145-156 (1991; Zbl 0735.51003)] \textit{P. Quattrocchi} and the author constructed an inversive plane from a Minkowski plane \({\mathcal M}\) as the geometry of points and blocks fixed by an automorphism \(\varphi\) of \({\mathcal M}\) that exchanges the two families of generators and fixes at least four points not on a block. The inversive plane obtained is then called \(\varphi\)-embedded into \({\mathcal M}\). In the paper under review the author continues her investigation of finite inversive planes \({\mathcal I}\) embedded (not necessarily \(\varphi\)-embedded) into Minkowski planes \({\mathcal M}\), that is, the point set of \({\mathcal I}\) is a subset of the point set of \({\mathcal M}\), circles of \({\mathcal I}\) are traces of blocks of \({\mathcal M}\) and tangency of circles in \({\mathcal I}\) is induced by the tangency of blocks in \({\mathcal M}\). Starting from a Miquelian Minkowski plane of order \(p^m\) the above construction for the automorphism \(\varphi:(x,y)\mapsto(y^{p^t},x^{p^t})\) is applied to obtain a \(\varphi\)-embedded inversive plane \(M(p^t)\) of order \(p^t\) for each positive integer \(t\) such that \(2t\) divides \(m\). (No \(\varphi\)-embedding can exist if \(m\) is odd; loc. cit.) It is then shown that, in fact, each such inversive plane \(M(p^t)\) is contained in each of the known finite Minkowski planes \({\mathcal M}(p^m,\sigma)\) of the same order \(p^m\), where \(\sigma\) is an automorphism of the Galois field GF\((p^m)\), and thus is embedded in \({\mathcal M}(p^m,\sigma)\). Furthermore, \(M(p^t)\) admits PSL\((2,p^{2t})\) as a group of automorphisms and hence must be Miquelian.