Subplanes of the Hughes plane of order 25
The inner structure is investigated of the Hughes plane of order 25, say \(\pi\). It is shown that \(\pi\) is singly generated by quadrangles with at most three real vertices, the real points of \(\pi\) being those in its real Baer subplane \(\pi _ 0\), i.e. the subplane fixed by all collineations of \(\pi\). In particular, a quadrangle with three real vertices generates one of the following: \((i) \pi;\) (ii) a Baer subplane with seven real points (lines), six of them on a line (point) which is referred to as an \(\alpha\)-Baer; (iii) a subplane of order three. Thus \(\pi\) provides the very first example of a plane of order \(p^ h\), p a prime, containing subplanes of order \(p'>2\), p' a prime, p'\(\neq p\). All subplanes of order three in \(\pi\) have three real points (dually, three real lines) and their distribution is discussed. Besides the \(\alpha\)-Baers, \(\pi\) contains Baer subplanes with six collinear (concurrent) real points (lines), the so-called \(\beta\)-Baers. Moreover, the Baer subplanes of \(\pi\) fall into exactly three transitivity classes under the full collineation group of \(\pi\), one of them consisting of \(\pi _ 0\) only. Some properties of the Baer subplanes of \(\pi\) are shown. Finally, the Fano subplanes are considered and it is proved that there are 21 inequivalent types of them. As a start, the subplanes of order two in \(\pi\) are classified to their first traces, i.e. the sets of their real points and lines. This yields eight inequivalent types. Such a classification is refined by looking at the sets of real points on the complex lines of the subplanes too.
scientific article; zbMATH DE number 3984001